Answer: 10 pencils and 5 notebooks.
Step-by-step explanation:
We will create a system of equations using the information given. Let n be equal to the number of notebooks and p be equal to the number of pencils.
She spent $23.25 on notebooks and pencils. Notebooks cost $2.49 each and pencils cost $1.08 each.
$2.49n + $1.08p = $23.25
She bought a total of 15 notebooks and pencils.
n + p = 15
Next, we will solve for p by substituting.
n + p = 15 ➜ n = 15 - p
$2.49n + $1.08p = $23.25
$2.49(15 - p) + $1.08p = $23.25
$37.35 - $2.49p + $1.08p = $23.25
$37.35 - $1.41p = $23.25
-$1.41p = -$14.10
p = 10 pencils
Lastly, we will solve for n by substituting:
n = 15 - p
n = 15 - 10
n = 5
A semi-elliptic archway has a height of 15 feet
at the center and a width of 50 feet, as shown
in the figure. The 50-foot width consists of a
two-lane road. Can a truck that is 12 feet high
and 14 feet wide drive under the archway
without going into the other lane?
Since 1.1584 > 1, the truck cannot pass under the archway without going into the other lane.
A semi-elliptic archway with a height of 15 feet at the center and a width of 50 feet can be visualized as half of an ellipse.
The major axis of the ellipse corresponds to the width, while the minor axis corresponds to the height. In this case, the major axis (a) is 25 feet, and the minor axis (b) is 15 feet.
To determine if a truck that is 12 feet high and 14 feet wide can drive under the archway without going into the other lane, we can use the equation of an ellipse: (x²/a²) + (y²/b²) = 1.
The truck will occupy half of the road width, which is 25 feet, so its horizontal distance from the center (x) is 25 - 7 = 18 feet, and its height (y) is 12 feet.
Plugging these values into the equation, we get: (18²/25²) + (12²/15²) = (324/625) + (144/225) ≈ 0.5184 + 0.64 = 1.1584.
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The library in natchitoches had about 60000 volumes in march 2004 and was adding about 400 volumes per month Write an equation expressing y, the total number of volumes in the library, in terms of x, the number of mouths after march 2004
The equation expressing y, as the total number of volumes in the library, in terms of x, the number of mouths after march 2004 is y = 400x + 60,000.
What is the equation for the volumes in library?The equation expressing y, as the total number of volumes in the library, in terms of x, the number of mouths after march 2004 is determined as follows;
Using general linear equation;
y = mx + b
where:
y is the total number of volumes in the libraryx is number of months after March 2004m is the rate of change b is the initial number of volumes in the library in March 2004The initial volume as at March = 60,000
the rate = 400 volumes / month
The equation is determined as;
y = 400x + 60,000
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Solve the right triangle. Round decimal answers to the nearest tenth.
G
14
H
?
J
16
HJ~
m angle G ~
m angle J~
The required values are HJ = 2√15 units, ∠G = 67.5 Degrees, ∠J = 61.04 degree.
What is Right angled triangle?A triangle with two sides that are perpendicular to one another is known as a right triangle, right-angled triangle, or orthogonal triangle. It was previously known as a rectangled triangle. Trigonometry is based on the relationship between the right triangle's sides and other angles.
According to question:Given data
GH = 14 units and GJ = 16 units
Using Pythagorean theorem;
[tex]16^2 = 14^2 + HJ^2[/tex]
[tex]HJ = \sqrt{16^2-14^2}[/tex]
HJ = √60
HJ = 2√15 units
And
Cos(G) = 14/16 = 7/8
∠G = cos⁻¹(7/8)
∠G = 67.5 Degrees
And
Sin(J) = 14/16 = 7/8
∠J = Sin⁻¹(7/8)
∠J = 61.04 degree
Thus, required values are HJ = 2√15 units, ∠G = 67.5 Degrees, ∠J = 61.04 degree.
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Jay has five fewer $100 bills in his wallet than $50 bills. If he has only $50 and $100 bills in his wallet, and the total amount of money in his wallet is $1300, let’s find the number of notes of each denomination.
Let’s assume the number of $50 bills as x. So, in terms of x, the number of $100 notes can be represented as ____.
Now, the total amount contributed by $50 notes is $50x
And. The total amount contributed by $100 bills is $100×( ____ )
Jay has 7 $100 bills in his wallet.
Let's assume the number of $50 bills as x. So, in terms of x, the number of $100 notes can be represented as x - 5.
Now, the total amount contributed by $50 notes is $50x.
And the total amount contributed by $100 bills is $100*(x-5).
Since the total amount of money in his wallet is $1300, we can write an equation:
$50x + $100*(x-5) = $1300
Simplifying this equation, we get:
$50x + $100x - $500 = $1300
$150x = $1800
x = 12
So, Jay has 12 $50 bills in his wallet.
Using the earlier equation, the number of $100 bills can be found:
x - 5 = 12 - 5 = 7
Therefore, Jay has 7 $100 bills in his wallet.
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Quadrilateral klmn is similar to quadrilateral opqr. find the measure of side op.
round your answer to the nearest tenth if necessary.
n
13
m
r
57
q
27
k
p
0
answer:
submit answer
The measure of side OP in quadrilateral OPQR is 0.
To find the measure of side OP in quadrilateral OPQR, which is similar to quadrilateral KLMN, follow these steps:
1. Identify the corresponding sides in both quadrilaterals. In this case, side OP corresponds to side KL, side OQ corresponds to side KM, side PQ corresponds to side LN, and side QR corresponds to side MN.
2. Determine the scale factor between the quadrilaterals by comparing the lengths of corresponding sides. Since we have the lengths of sides KM (13), LN (27), and MN (57), we can use the ratio of KM/LN (13/27) or MN/LN (57/27) as the scale factor.
3. Apply the scale factor to the length of side KL (0) to find the length of side OP. Since the length of side KL is 0, multiplying by the scale factor (either 13/27 or 57/27) will still result in a length of 0 for side OP.
4. Round your answer to the nearest tenth if necessary. In this case, the length of side OP is 0, so rounding is not necessary.
So, the measure of side OP in quadrilateral OPQR is 0.
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12 Let f(x)= x²/4(x+1) Find all critical numbers of f. As your answer please input the sum of all critical numbers.
The critical numbers of f(x) are x = -1, 0, and 1 and The sum of all critical numbers is 0.
How to find the critical numbers?To find the critical numbers of f(x) = x²/4(x+1), we need to find the values of x where the derivative of f(x) is equal to zero or does not exist.
The derivative of f(x) is:
f'(x) = [(x+1)(2x) - x²(4)] / [4(x+1)²]
Simplifying, we get:
f'(x) = [2x(x+1) - 4x²] / [4(x+1)²]
f'(x) = [2x(x+1-2x)] / [4(x+1)²]
f'(x) = [2x(1-x)] / [4(x+1)²]
f'(x) = [x(1-x)] / [2(x+1)²]
The critical numbers are the values of x where f'(x) is equal to zero or does not exist.
Setting f'(x) = 0, we get:
x(1-x) = 0
This equation is true when x = 0 or x = 1.
Now, let's check if f'(x) does not exist at x = -1 (which is the only possible point where the derivative may not exist):
f'(x) = [x(1-x)] / [2(x+1)²]
When x = -1, the denominator of f'(x) becomes zero, so the derivative does not exist at x = -1.
Therefore, the critical numbers of f(x) are x = -1, 0, and 1.
The sum of all critical numbers is:
-1 + 0 + 1 = 0
Hence, the answer is 0.
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Keshia is working on a math worksheet with 50 problems she has completed 20 problems in 25 mins if she continue at this pace what will be her total time for her compete worksheet
It will take Keshia 62.5 minutes to complete the entire worksheet at the same pace.
To find the amount of total time it will take Keshia to complete the entire worksheet at the same pace, we can use a proportion:
20 problems / 25 minutes = 50 problems / x minutes
To solve for x, we can cross-multiply and simplify:
20 problems * x minutes = 25 minutes * 50 problems
20x = 1250
x = 1250 / 20
x = 62.5
Therefore, it will take Keshia 62.5 minutes to complete the entire worksheet at the same pace.
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Your doing practice 3
Based on the information, the three numbers are 14, 34, and 70.
What are the numbers?Based on the information, the second number = 3x - 8
The third number is five times the first number, which can be written as:
third number = 5x
The sum of the three numbers is 118, so we can write an equation:
x + (3x - 8) + 5x = 118
9x - 8 = 118
Adding 8 to both sides:
9x = 126
x = 236 / 914
Now we can use this value of x to find the other two numbers:
second number = 3x - 8 = 3(14) - 8 = 34
third number = 5x = 5(14) = 70
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find the number of triangles that can be formed using 14 points in a plane such that 4 points are collinear?
The number of triangles that can be formed using 14 points in a plane such that 4 points are collinear is 360.
If 4 points are collinear, then we can choose any 3 of those points to form a line segment, which cannot be extended to form a triangle.
we need to choose 3 points from the 14 points such that no 3 of them are collinear.
The number of triangles that can be formed from n non-collinear points in a plane is given by the formula:
C(n,3) = n(n-1)(n-2)/6
where C(n,3) is the binomial coefficient for choosing 3 items out of n.
So, to find the number of triangles that can be formed using 14 points in a plane such that 4 points are collinear,
we first need to find the number of ways to choose 3 points from 14 points, and then subtract the number of ways to choose 3 points from the 4 collinear points.
That is:
Total number of triangles = C(14,3) - C(4,3)
Total number of triangles = 364 - 4
Total number of triangles = 360
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An investor is planning on selling some property that she recently purchased. A real estate consulting firm determines that there is a 50% chance of making a profit of $50,000, a 30% chance of breaking even, and a 20% chance of suffering a $60,000 loss. Determine the expected value of the sale
The expected value of the sale is $13,000.
How to determine the expected value of the sale?The expected value is a statistical measure that represents the average outcome of a probability distribution, weighted by the probabilities of each outcome. In this case, the investor is planning to sell a property and wants to know what the expected value of the sale will be. To determine this value, we must consider the potential outcomes and their probabilities.
According to the real estate consulting firm, there is a 50% chance of making a profit of $50,000, a 30% chance of breaking even, and a 20% chance of suffering a $60,000 loss. To calculate the expected value of the sale, we multiply the potential profit or loss by the probability of each outcome occurring and then sum those products.
To determine the expected value of the sale, we need to multiply the potential profit or loss by the probability of each outcome occurring and then sum those products.
Expected value = (0.5 * $50,000) + (0.3 * $0) + (0.2 * -$60,000)
Expected value = $25,000 - $12,000
Expected value = $13,000
Therefore, the expected value of the sale is $13,000.
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Given that 3 is a primitive root modulo 25; Find a primitive root modulo 250
The primitive root modulo 250 is 103, as 3 is a primitive root modulo 25 we tested if it is also a primitive root modulo 250.
To find a primitive root modulo 250, we need to first factor 250 as 2 x 5³. Since 3 is a primitive root modulo 25, we can test if it is also a primitive root modulo 250.
Using Euler's totient function, we know that [tex]\phi(250)[/tex] = 100. Therefore, we only need to check if [tex]3^{20[/tex] (which is [tex]3^{\phi(250)/2[/tex]) is congruent to -1 modulo 250.
Calculating [tex]3^{20[/tex] modulo 250 gives us 1. Since [tex]3^{20[/tex] is not congruent to -1 modulo 250, 3 is not a primitive root modulo 250.
To find a primitive root modulo 250, we can use a common method called the "index cycling" method. We can start with a primitive root modulo 5³ = 125 and then test the other primitive roots modulo 2³ = 8 until we find a primitive root modulo 250.
Using a computer or calculator, we can find that 2 is a primitive root modulo 125. To find a primitive root modulo 250, we can test the numbers 2, 2 + 125, 2 + 2125, and 2 + 3125 until we find a primitive root.
Testing these numbers, we find that 2 + 3*125 = 377 is a primitive root modulo 250. Therefore, 377 is a primitive root modulo 250.
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1 point
6. The radius of the circular garden pond is 1. 75 feet. If a landscaper wants
to place a decorative fence around the circumference of the pond, about
how many feet of fencing will be needed? *
O 1. 099 feet
O 10. 99 feet
109. 9 feet
O 10. 99 square feet
O 1,099 square feet
The landscaper will need 10.99 feet of fencing to place around the circumference of the pond. Option B is the correct answer.
We need to find how many feet of the fence is needed to decorate the fence around the circumference of the pond. we can determine it by finding the circumference of a pound or circle. The circumference of a circle is calculated using the formula,
C = 2πr
Where:
C = the circumference
r = radius
Given data:
π = 3.14
r = 1. 75 feet.
Substuting the value of the radius in the formula we get
C = 2πr
C = 2π(1.75)
= 10.99
Therefore, the landscaper will need 10.99 feet of fencing to place around the circumference of the pond.
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Which of the following is a counterexample for the following statement?
“If a line intersects a circle, then it intersects it in two points.”
According to the information, and example of a counterexample is: A line can cross the circle only at one point.
What would be a counterexample for this statement?To find a counterexample to this statement we must read it carefully and identify the main idea of it. In this case you are stating that a line always crosses a circle at two points.
Later we must analyze this statement and evaluate if it is true or false. In this case it is false because we can make a line that crosses a circle only once. So, the counter example sentence would be.
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Determine the intercepts of the line.
Do not round your answers.
5x − 9 = −8y − 3
Please help
The intercepts of the line are (6/5, 0) and (0, -3/4).
We have,
To find the x-intercept, we need to set y = 0 and solve for x:
5x - 9 = -3
5x = 6
x = 6/5
So the x-intercept is (6/5, 0).
To find the y-intercept, we need to set x = 0 and solve for y:
-9 = -8y - 3
8y = -6
y = -3/4
So the y-intercept is (0, -3/4).
Therefore,
The intercepts of the line are (6/5, 0) and (0, -3/4).
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What is the circumference of the following circle?
Use 3.14 for πpi and enter your answer as a decimal.
The calculated value of the circumference of the circle is 31.4 units
What is the circumference of the following circle?From the question, we have the following parameters that can be used in our computation:
Radius, r = 5
Using the above as a guide, we have the following:
Circumference = 2 * π * r
Substitute the known values in the above equation, so, we have the following representation
Circumference = 2 * 5 * 3.14
Evaluate the products
Circumference = 31.4
HEnce, the value of the circumference is 31.4 units
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y = 49 - 4x y = 73 - 7x
The solution of the system of equation
x = 8 and y = 17
How to solve system of equation?System of equation can be solved using different method such as elimination method, substitution method and graphical method. Let's solve the system of equation by substitution method.
Therefore,
y = 49 - 4x
y = 73 - 7x
Hence, using substitution,
73 - 7x = 49 - 4x
73 - 49 = -4x + 7x
24 = 3x
divide both sides of the equation by 3
x = 24 / 3
x = 8
Therefore,
y = 73 - 7(8)
y = 73 - 56
y = 17
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Find the measure of the missing angle
38°
X
Y
The measure of angle x is 77 degrees.
How to calculate the angleWe have two angles given: 65 degrees and 38 degrees. Let's call the measure of angle x as "x".
From the information, we have the measure of the missing angle of the angles in a triangle are 38°, 65° and x. Then we can set up an equation:
x + 65 + 38 = 180 (the sum of the measures of the angles in a triangle is 180)
Simplifying this equation, we get:
x + 103 = 180
x = 77
Therefore, the measure of angle x is 77 degrees.
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Complete question
Find the measure of the missing angle of the angles in a triangle are 38°, 65° and x.
Suppose an amusement park is being built in a city with a population of 100. Voluntary contributions are being solicited to cover the cost. Each citizen is being ask to give $100. The more people contribute, the larger the park will be and the greater the benefit to each citizen. But it is not possible to keep out the noncontributors; they get their share of this benefits anyway. Suppose that when there are n contributors in the population, where n can be any whole number between 0 and 100. The benefit to each citizen in monetary unit equivalents in n 2 dollars.
Required:
a. Suppose that initially no one is contributing. You are the mayor of the city. You would like everyone to contribute and can use persuasion on some people. What is the minimum number whom you need to persuade before everyone else will join voluntarily?
b. Find the Nash equilibria of the game where each citizen is deciding whether to contribute
The minimum number of people that need to be persuaded is two. When there are 0 contributor, 1 contributor, 2 or more contributor this is a the Nash equilibria.
a. Let's first calculate the benefit to each citizen when there are n contributors. According to the problem, the benefit is n^2 dollars. So when there are 0 contributors, the benefit to each citizen is 0 dollars. When there is 1 contributor, the benefit to each citizen is 1 dollar. When there are 2 contributors, the benefit to each citizen is 4 dollars. And so on, up to 10,000 dollars per citizen when all 100 citizens contribute.
Now let's think about the incentives of each citizen to contribute. If no one contributes, everyone gets 0 dollars of benefit. If one person contributes, that person gets 1 dollar of benefit, and everyone else gets 0 dollars. So each person has an incentive to free-ride, hoping that someone else will contribute.
But if two people contribute, each person gets 4 dollars of benefit, which is more than the 1 dollar cost of contributing. So once there are at least two contributors, it becomes rational for everyone else to contribute as well.
Therefore, the minimum number of people that need to be persuaded is two. Once two people contribute, it becomes rational for everyone else to contribute as well.
b. Let's consider the Nash equilibria of the game where each citizen is deciding whether to contribute. A Nash equilibrium is a situation where no one has an incentive to change their strategy, given the strategies of all the other players.
In this case, each citizen has two strategies: contribute or free-ride. Let's consider the case where n citizens are contributing. If everyone else is contributing, then it is rational to contribute as well, since the benefit of contributing is greater than the cost.
If everyone else is free-riding, then it is rational to free-ride as well, since the cost of contributing is greater than the benefit. However, if some people are contributing and some people are free-riding, then it may be rational to contribute, since the benefit of contributing may outweigh the cost, depending on the number of contributors.
Therefore, there are multiple Nash equilibria in this game, depending on the number of contributors. When there are 0 contributors, everyone is free-riding and this is a Nash equilibrium. When there is 1 contributor, that person is contributing and everyone else is free-riding, and this is a Nash equilibrium. When there are 2 or more contributors, everyone is contributing, and this is a Nash equilibrium.
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a repair shop charges per hour the amount of money they charge is below what equation does the repair shop use.
1 $70 Here are the answers i can chose from
c= 30h+ 40 C= 40h+30 C=30h+70 C= 40h
h is for hours and c is for charge
2 $100
3 $130
4 $160
5 $190
6 $220
The linear function for the repair cost is given as follows:
C(h) = 30h + 40.
How to define a linear function?The slope-intercept representation of a linear function is given by the equation shown as follows:
y = mx + b
The coefficients m and b have the meaning presented as follows:
m is the slope of the function, representing the increase/decrease in the output variable y when the input variable x is increased by one.b is the y-intercept of the function, representing the numeric value of the function when the input variable x has a value of 0. On a graph, it is the value of y when the graph of the function crosses or touches the y-axis.Each hour the cost increases by $30, hence the slope m is given as follows:
m = 30.
Hence the function is:
C(h) = 30h + b
When h = 2, C = 100, hence the intercept b is given as follows:
60 + b = 100
b = 40.
Then:
C(h) = 30h + 40.
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Determine the length of the interior bathroom wall(excluding the door) that is not goven if the door takes a take space of 860mm 2.The kitchen and the bathroom should be tiled .The floor tile dimension is 500mm by 500mm .when purchasing tiles you need to buy 5% more to cater for breakages .A tiling company charges R 8180.00 for labour and can supply the tiles for R 249.00 per box NOTE::area=l×width ..all items like the bath ,stives,cupboard are movable items and tiling will be done on the spaces where they will be placed 1.calculate the total area that must be tiled in metres (length=6030mm inner dimension excluding the bedroom but also calculate it and outer is 12330 mm and width =4680mm and 5130 mm excluding the bath area outer is 13680mm 3.2.2 the building manager made a statement that 150 tiles are needed to complete the tiling for the kitchen and bathroom .verify with calculations whether this statement is valid or not(Length=6030mm width=5130 mm for kitchen....bathroom =l 2250 mm width =13680 outer dimension including 4680 mm for bedroom 1 and 5130 mm for bedroom 2
A total number of 59.6001 tiles (approximately 60 tiles) are needed to complete the tiling for the kitchen and bathroom.
To calculate the total area that needs to be tiled, we'll start by converting the given dimensions from millimeters to meters:
Bathroom Inner Dimensions:
Length = 6030 mm = 6.03 m
Width = 5130 mm = 5.13 m
Bathroom Outer Dimensions (including bedroom areas):
Length = 12330 mm = 12.33 m
Width = 4680 mm = 4.68 m
Kitchen Dimensions:
Length = 6030 mm = 6.03 m
Width = 5130 mm = 5.13 m
Total area to be tiled in the bathroom (excluding the bath area):
Area = Length x Width = 6.03 m x (5.13 m - 0.86 m) = 6.03 m x 4.27 m = 25.7701 m²
Total area to be tiled in the kitchen:
Area = Length x Width = 6.03 m x 5.13 m = 30.9919 m²
Total area to be tiled (bathroom + kitchen):
Total Area = 25.7701 m² + 30.9919 m² = 56.762 m²
To account for breakages, we need to purchase 5% more tiles. So, the total number of tiles needed is:
Total Number of Tiles = Total Area x 1.05 (to account for 5% extra)
Total Number of Tiles = 56.762 m² x 1.05 = 59.6001 tiles
The building manager stated that 150 tiles are needed. Comparing this with our calculation:
150 tiles < 59.6001 tiles
Therefore, the statement made by the building manager is not valid. According to our calculations, a total of 59.6001 tiles (approximately 60 tiles) are needed to complete the tiling for the kitchen and bathroom.
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Find the slope of the line through points 6,3 and 12,7
The slope of the line through points (6,3) and (12,7) is 2/3.
To find the slope of a line, we use the formula:
Slope = (y2 - y1) / (x2 - x1)
In this case, we have two points: (6, 3) and (12, 7). We can label them as follows:
x1 = 6
y1 = 3
x2 = 12
y2 = 7
Now we can plug these values into the formula:
Slope = (y2 - y1) / (x2 - x1)
Slope = (7 - 3) / (12 - 6)
Slope = 4 / 6
Slope = 2/3
Therefore, the slope of the line through the points (6, 3) and (12, 7) is 2/3.
The slope of a line tells us how steep it is. A positive slope means the line goes up as you move from left to right, while a negative slope means the line goes down. In this case, since the slope is positive (2/3), we know that the line goes up as we move from left to right.
The slope also tells us how much the y-value changes for every one unit of x-value. In this case, for every one unit we move to the right, the y-value goes up by 2/3.
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A right rectangular prism has a length of f 2 feet, a width of 3 feet, and a height of
1
1/2/14 feet. Unit cubes with side lengths of foot are added to completely fill the prism
with no space remaining. What is the volume, in cubic feet, of the right rectangular
prism?
Show your work.
The volume, in cubic feet, of the right rectangular prism is 9 cubic feet
What is the volume, in cubic feet, of the right rectangular prism?From the question, we have the following parameters that can be used in our computation:
A right rectangular prism has a length of 2 feet, a width of 3 feet, and a height of 1 2/4 feet
Using the above as a guide, we have the following:
Volume = Length * Width * height
Substitute the known values in the above equation, so, we have the following representation
Volume = 2 * 3 * (1 2/4)
Evaluate
Volume = 9
Hence, the volume is 9 cubic feet
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If Sarah uses 3/4 yard of ribbon to make a hair bow. How many yards of ribbon will Sarah use to make 9 hair bows?
If Sarah uses 3/4 yard of ribbon to make a hair bow, she will need 6 and 3/4 yards of ribbon to make 9 hair bows.
To find out how many yards of ribbon Sarah will use to make 9 hair bows, we need to multiply the amount of ribbon used for one hair bow (3/4 yard) by the number of hair bows she wants to make (9).
So, the equation we need to use is:
3/4 yard of ribbon per hair bow x 9 hair bows = ? yards of ribbon
To solve for the answer, we can simplify the equation:
3/4 x 9 = 27/4
So Sarah will need 27/4 yards of ribbon to make 9 hair bows.
To convert this fraction to a mixed number, we can divide the numerator (27) by the denominator (4) and write the remainder as a fraction:
27 ÷ 4 = 6 with a remainder of 3
In summary, Sarah will need 6 and 3/4 yards of ribbon to make 9 hair bows, if she uses 3/4 yard of ribbon to make one hair bow.
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The function f(x)=8x+3x^−1 has one local minimum and one local maximum.
This function has a local maximum at x= With a value of =
This function has a local minimum at x = With a value of =
The function f(x) = 8x + 3x^(-1) has a local maximum at x = 2 with a value of f(2) = 19, and a local minimum at x = -2 with a value of f(-2) = -19.Explanation:
To find the local extrema of a function, we need to find the critical points of the function, which are the points where the derivative is either zero or undefined. In this case, the derivative of f(x) is f'(x) = 8 - 3x^(-2), which is undefined at x = 0.Setting the derivative equal to zero, we get:8 - 3x^(-2) = 0Solving for x, we get:x = ±2
These are the critical points of the function. To determine whether each critical point is a local maximum or a local minimum, we need to examine the second derivative of the function.
The second derivative of f(x) is f''(x) = 6x^(-3), which is negative for x > 0 and positive for x < 0.Therefore, x = 2 is a local maximum of the function with a value of f(2) = 19, and x = -2 is a local minimum of the function with a value of f(-2) = -19. These are the only local extrema of the function, since the function is increasing for x < -2 and decreasing for -2 < x < 0, and then increasing again for x > 0.
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Please Help. All I have been getting is bot answers, and I actually need help with this. Thanks.
1. Two neighbors are each hosting a party. The first neighbor orders 5 large pizzas, each with a diameter of 16 inches. The second neighbor orders 9 small pizzas, each with a diameter of 12 inches. In terms of area, which party has more pizza? Explain. Show all work.
2. A toy train set has a circular track piece. The inner radius of the piece is 6 cm. One sector of the track has an arc length of 33 cm on the inside and 55 cm on the outside. What is the width of the track?
1. The second party has more pizza in terms of area
2. The width of the track is approximately 3.50 cm.
How to find which party has more pizza?1. To compare the amount of pizza between the two parties, we need to find the total area of each set of pizzas. We can use the formula for the area of a circle, A = πr², where r is the radius of the circle.
For the large pizzas, the diameter is 16 inches, so the radius is 8 inches. The area of one pizza is:
A = π(8)²
A = 64π square inches
Since there are 5 pizzas, the total area of pizza for the first party is:
Total area = 5(64π) = 320π square inches
For the small pizzas, the diameter is 12 inches, so the radius is 6 inches. The area of one pizza is:
A = π(6)²
A = 36π square inches
Since there are 9 pizzas, the total area of pizza for the second party is:
Total area = 9(36π) = 324π square inches
Therefore, the second party has more pizza in terms of area.
How to find width of the track?2. We can start by finding the length of the arc of the sector. We know that the arc length on the inside is 33 cm, so the angle of the sector can be found using the formula:
angle = (arc length / radius)
angle = (33 / 6) radians
To find the width of the track, we need to subtract the length of the inner circle from the length of the outer circle, and then divide by the angle of the sector. Let's call the width of the track "x". Then we have:
55 cm - 33 cm = 2π(6 cm + x) - 2π(6 cm)
22 cm = 2πx
x = 11 / π cm
So the width of the track is approximately 3.50 cm.
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measured in astronomical units, can be modeled using the expression ((1)/(52)x)^((2)/(3)) , where x is the number of Earth weeks it takes for the planet to orbit the sun. Which expression could also be used to represent the average distance of a planet from the sun using radicals?
So the expression that represents the average distance of a planet from the sun using radicals is: d = k/2√13 * √x
What is exponent?An exponent, also known as a power, is a mathematical notation that indicates the number of times a quantity is multiplied by itself. It is usually written as a small number (the exponent) placed to the right and above a larger number (the base). Exponents are used in many mathematical concepts, including logarithms, roots, and scientific notation.
Here,
The expression ((1)/∛(52)x)²) can be simplified using exponent rules:
((1)/∛(52)x)²) =((1)/∛(52)x)²) * ∛x²)
= 1/(∛52² * ∛x²)
The average distance of a planet from the sun measured in astronomical units can be represented using the formula:
d = k * √T
where d is the distance from the sun, T is the time it takes for the planet to orbit the sun, and k is a constant of proportionality.
We can rewrite this formula in terms of Earth weeks by noting that there are 52 weeks in a year, so T = (1/52)x years. Substituting this into the formula, we get:
d = k * √((1/52)x)
Simplifying this expression using exponent rules, we get:
d = k * √(1/52)* √x
So an equivalent expression using radicals to represent the average distance of a planet from the sun is:
d = k * √(1/(52)) * √x
which simplifies to:
d = k/√(52) * √x
or
d = k/2√13 * √x
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A rectangular prism has a volume of 27 in³ If a rectangular pyramid has a base and height congruent to the prism, what is the volume of the pyramid?
___in³
fill in the blank
ty
A rhombus has a perimeter of 88 and
two acute angles that measure 40° each.
Find the length of the shorter diagonal.
PLEASE HELP ME ASAP
The length of the shorter diagonal is approximately 7.07 units.
A rhombus is a four-sided polygon in which all sides are equal in length. It is also called a diamond shape because it is often used for diamond-shaped figures. In this case, we are given that the perimeter of the rhombus is 88. Since all sides of the rhombus are equal, we can divide the perimeter by 4 to find the length of one side.
88 ÷ 4 = 22
Therefore, each side of the rhombus measures 22 units.
We are also given that two of the angles in the rhombus are acute angles measuring 40° each. Since all angles in a rhombus are equal, we can find the measure of the other two angles by subtracting the sum of the acute angles from 360.
360 - 2(40) = 280
Each of the other two angles measures 140°.
To find the length of the shorter diagonal, we can use the formula:
Shorter diagonal = (2 × Area) / Length of longer diagonal
The area of a rhombus can be found by multiplying the length of the longer diagonal by the length of the shorter diagonal and then dividing by 2.
Area = (diagonal1 × diagonal2) / 2
We know that the longer diagonal is twice the length of the shorter diagonal.
Longer diagonal = 2 × Shorter diagonal
Substituting these values into the formula, we get:
Shorter diagonal = (2 × (22 × Shorter diagonal × sin 40°)) / (2 × Longer diagonal)
Simplifying, we get:
Shorter diagonal = (22 × Shorter diagonal × sin 40°) / Longer diagonal
Plugging in the values we know, we get:
Shorter diagonal = (22 × Shorter diagonal × sin 40°) / (2 × Shorter diagonal)
Shorter diagonal = 11 × sin 40°
Using a calculator, we can find that:
Shorter diagonal ≈ 7.07
Therefore, the length of the shorter diagonal is approximately 7.07 units.
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Ion
theodora opens a savings account with an initial deposit of $120. he then deposits $120 into that
savings account at the end of every subsequent month. this savings account pays an annual interest
rate of 2.9% and is compounded monthly. how much does keelan have in his account at the end of
3 years? round your answer to the nearest penny.
Ion has a total of $4,378.05 in his savings account at the end of 3 years.
To solve this problem, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the total amount of money in the account at the end of the time period
P = the principal amount (initial deposit)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the time period (in years)
In this case, we have:
P = $120
r = 0.029 (2.9% expressed as a decimal)
n = 12 (compounded monthly)
t = 3 (years)
We also need to calculate the total number of deposits made during the 3-year period. Since a deposit of $120 is made at the end of every month, and there are 12 months in a year, the total number of deposits made is:
12 deposits/year × 3 years = 36 deposits
Now we can plug in the values into the formula:
A = [tex]120(1 + 0.029/12)^(12 × 3) + 120[(1 + 0.029/12)^(12 × 3) - 1]/(0.029/12)[/tex]
A = $4,378.05
Therefore, Ion has a total of $4,378.05 in his savings account at the end of 3 years.
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Researchers pose a question can pant sizes be predicted from a man's height? A random sample of 20 males and their pant size versus height
Answer:
Yes, there is evidence at the 10% significance level, but not at the 5% level
Step-by-step explanation:
Look at the p-value in bottom left corner (in the Height row).
It is less than 0.1, but greater than 0.05. Thus, Yes, there is evidence at the 10% significance level, but not at the 5% level. Got it right.
