Polar coordinates for rectangular coordinates if x<0: r=√(x²+y²) and θ=tan⁻¹(y/x)+π if y≥0 or θ=tan⁻¹(y/x)−π if y<0, For x≥0: r=√(x²+y²) and θ=tan⁻¹(y/x) if y≥0 or θ=tan⁻¹(y/x)+2π if y<0.
The polar coordinates of a point with rectangular coordinates (x,y) depend on the sign of x.
If x<0, the polar coordinates are (r,θ) where r≥0 and −π/2≤θ<3π/2. If x≥0, the polar coordinates are (r,θ) where r≥0 and −π/2≤θ<3π/2.
If x<0, t
hen r=√(x²+y²) and
θ=tan⁻¹(y/x)+π if y≥0
or θ=tan⁻¹(y/x)−π if y<0.
The value of r is the distance from the origin to the point and θ is the angle between the positive x-axis and the line segment from the origin to the point.
If x≥0, then r=√(x²+y²) and θ=tan⁻¹(y/x) if y≥0 or θ=tan⁻¹(y/x)+2π if y<0.
In this case, θ is the angle between the positive x-axis and the line segment from the origin to the point, measured counterclockwise.
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A rectangular field is 63 yards long and 21 yards wide. A fence is needed for the perimeters of the field. Fencing is also needed to divide the field into three square sections. How many feet of fencing are needed? Show step-by-step.
Answer: 210 yards of fencing will be needed
Step-by-step explanation: well the perimeter of this rectangular field is 21 + 63 + 21 + 63 yards or 2(21) + 2(63) yards which equals 168 yards.
To divide the field into 3 equal parts, u need to divide the length (63 yards) into 3 parts which also requires two more lines of fencing.
63/3=21 which means u get squares perfect squares when u divide. now that means that's an additional 21*2 yards of fencing since you need two more rows of fencing in the middle of the field to divide the length into three equal parts. 21*2 = 42 so thats an additional 42 yards. The total amount of fencing is 168 + 42 = 210 yards.
The circumstances of the base the cone is 60π cm. If the volume of the cone is 21,600π cm cubed, what is the height?
Answer:
h = 24 cm
Step-by-step explanation:
Given:
C (base) = 60π cm
V (volume) = 21,600π cm^3
Find: h (height) - ?
[tex]c = 2\pi \times r[/tex]
[tex]2\pi \times r = 60\pi[/tex]
[tex]2r = 60[/tex]
[tex]r = 30[/tex]
We found the length of the radius
v = 1/3 × πr^2 × h
1/3 × π × 900 × h = 21600π
Multiply both sides by 3:
2700π × h = 64800π / : 2700π
h = 24 cm
10- 4x + 6 - 2x = -2x
Answer:
x = 4
Step-by-step explanation:
10 - 4x + 6 - 2x = -2x
10 - 6x + 6 = -2x
16 - 6x = -2x
16 - 4x = 0
-4x = -16
x = 4
Answer:
x = 4
Step-by-step explanation:
Add like terms
-6x + 16 = -2x
Bring like terms to the opposite side
16 = 4x
Divide both sides by 4
x = 4
Let f(x) = x^2 (Inx-1). (a) Find the critical numbers of f. (b) Find the open interval(s) on which f is increasing and the open interval(s) on which f is decreasing. (c) Find the local minimum value(s) and local maximum value(s) off. if any. (d) Find the open interval(s) where f is concave upward and the open interval(s) where f is concave downward. (e) Find the inflection point(s) of the graph of f, if any.
a. The critical number of f is undefine
b. The open interval(s) on f is increasing on (e,∞) and the open interval(s) on which f is decreasing on (0,1) and (1,e).
c. The local minimum value(s) is 0 and there's no local maximum value.
d. Concave downward on (0, e^1/2) and concave upward on (e^1/2, ∞).
e. The inflection point(s) of the graph of f is (e^1/2, e(ln e^1/2 - 1)^2).
(a) To find the critical numbers of f, we need to find where the derivative of f is zero or undefined.
f'(x) = 2x ln x + x - 2x = 2x (ln x - 1) = 0
This gives us x = 1 or x = e. However, f'(x) is undefined at x = 0, so we also need to check this point.
(b) To determine the intervals of increase and decrease, we need to test the sign of f'(x) on each interval.
When x < 1, ln x < 0, so ln x - 1 < -1, and f'(x) < 0.
When 1 < x < e, ln x > 0, so ln x - 1 < 0, and f'(x) < 0.
When x > e, ln x > 1, so ln x - 1 > 0, and f'(x) > 0.
Therefore, f is decreasing on (0,1) and (1,e), and increasing on (e,∞).
(c) To find the local minimum and maximum values, we need to check the critical points and the endpoints of the intervals.
f(1) = 0 is a local minimum.
f(e) = e^2 (ln e - 1) = e^2 (1 - 1) = 0 is also a local minimum.
(d) To find the intervals of concavity, we need to test the sign of f''(x) on each interval.
f''(x) = 2 ln x - 1
When x < e^1/2, ln x < 1/2, so f''(x) < 0, and f is concave downward on (0, e^1/2).
When x > e^1/2, ln x > 1/2, so f''(x) > 0, and f is concave upward on (e^1/2, ∞).
(e) To find the inflection points, we need to find where the concavity changes.
f''(x) = 0 when ln x = 1/2, or x = e^1/2.
Therefore, the inflection point is (e^1/2, f(e^1/2)) = (e^1/2, e(ln e^1/2 - 1)^2).
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What is the midpoint of the line segment that joins points (4,-2) and (-2,5)
Answer:
(1, 1.5).
Step-by-step explanation:
Midpoint of (x1, y1) and (x2, y2)
(x1 + x2)/2 , (y1 + y2)/2
=(4+-2)/2, (-2+5)/2
= (1, 1.5).
How do I find the value of X
Answer:
x = 30
Step-by-step explanation:
you can find it by vertical opposite angle which is their non adjecent angle are equal.
2x- 30 = 30
2x = 30 + 30 ..... take 30 to the left it change sighn
2x /2 = 60 / 2 ..... divided both side by 3
x = 30 ..... so the unknown no. is 30
What is the cosine ratio for angle A ?
A. 6/10
B. 6/8
C. 6/10
D. 8/10
Answer:
8/10
Step-by-step explanation:
Formula
Cosine A = Adjacent side / Hypotenuse
Here,
Adjacent side = 8
Hypotenuse = 10
Answer
Cosine A = 8/10
Given :f(x)=∑ n=1 ∞ (x+2) Determine: the values of x for which f(x) converges . the value of (x) if x = 1/ 1/2
a. In the geometric series for f(x) to be convergent, x < - 1
b. When x = 1¹/₂, the sum to infinity of the geometric series is f(x) = -1.4
What is a geometric series?A geometric series is the sum of terms of a geometric sequence.
a. Given the series f(x) = ∑ₙ = ₁⁰⁰(x + 2)ⁿ, we want to determine the value of x for which f(x) converges.
Now, let the general term of the sequence be Uₙ = (x + 2)ⁿ, to determine the value of x for which the series is convergent, we use the D'alembert ratio test which states that for a series to be convergent, then
Uₙ₊₁/Uₙ < 1.
So, we have that Uₙ₊₁ = (x + 2)ⁿ⁺¹
So, Uₙ₊₁/Uₙ = (x + 2)ⁿ⁺¹/ (x + 2)ⁿ
= x + 2
For convergence
Uₙ₊₁/Uₙ < 1
So,
x + 2 < 1
x < 1 - 2
x < - 1
So, for f(x) to be convergent, x < - 1
b. To find the value of f(x) when x = 1¹/₂, we proceed as folows
Since f(x) = ∑ₙ = ₁⁰⁰(x + 2)ⁿ, substituting x = 1¹/₂ = 1.5 into the equation, we have
f(x) = ∑ₙ = ₁⁰⁰(1.5 + 2)ⁿ
f(x) = ∑ₙ = ₁⁰⁰(3.5)ⁿ
= 3.5 + 3.5² + 3.5³ + ...
Since this is a geometric progression with sum to infinity, we see that the first term is a = 3.5 and the common ratio is r = ar/a = 3.5²/3.5 = 3.5
Since the sum to infinity of a geometric progression is
S₀₀ = a/(1 - r)
So, substituting the values of the variables into the equation, we have that
S₀₀ = a/(1 - r)
S₀₀ = 3.5/(1 - 3.5)
S₀₀ = 3.5/-2.5
S₀₀ = -1.4
So, when x = 1¹/₂, f(x) = -1.4
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Find the value(s) of the variable(s). if necessary, round decimal answers
to the nearest tenth.
To find the value(s) of the variable(s), you need to have an equation or problem statement that relates the variable(s) to other known quantities. Once you have this equation or statement, you can solve for the variable(s) by manipulating the equation algebraically.
For example, if the problem states that 2x + 5 = 17, you can solve for x by first subtracting 5 from both sides to get 2x = 12. Then, you can divide both sides by 2 to get x = 6. So, the value of the variable x is 6.
In some cases, you may need to use more advanced methods such as factoring or the quadratic formula to solve for the variable(s). Regardless of the method used, it's important to check your answer(s) by plugging them back into the original equation to make sure they satisfy the given conditions.
In terms of rounding decimal answers to the nearest tenth, this means that if the answer is a decimal with more than one digit after the decimal point, you would round to the nearest tenth place (i.e. the digit immediately to the right of the decimal point). For example, if the answer is 3.456, you would round to 3.5.
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Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form Passing through (-9,2) and parallel to the line whose equation is y = – 3x + 3
The equation of the line in point-slope form is y - 2 = -3(x + 9), and in slope-intercept form is y = -3x - 25.
To find the equation of the line passing through(- 9,2) and parallel to the line y = – 3x 3, we need to use the fact that resemblant lines have the same pitch.
The pitch of the given line y = – 3x 3 is-3,
so the pitch of the resemblant line we want to find is also-3. Point- pitch form Using the point- pitch form, the equation of the line is given by
y- y1 = m( x- x1),
where( x1, y1) is the given point and
m is the pitch.
Substituting the given values,
we get y- 2 = -3( x-(- 9))
y- 2 = -3( x 9)
y- 2 = -3 x- 27
y = -3 x- 25
Hence in slope-intercept form is y = -3x - 25.
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For the function f (x) x = 2 to r 2 to X = 2.001. . x3, find the slope of secant over the interval A. slope = 10. 001001 B. slope = 1.006001 C. slope 2. 001001 D. slope - 12. 006001
To find the slope of the secant over the interval from x=2 to x=2.001, we need to use the formula for the slope of a secant line:
slope = (f(2.001) - f(2)) / (2.001 - 2)
First, we need to find the values of f(2.001) and f(2):
f(2) = 2^3 = 8
f(2.001) = 2.001^3 ≈ 8.012006001
Plugging these values into the formula, we get:
slope = (8.012006001 - 8) / (2.001 - 2)
slope ≈ 1.006001
Therefore, the slope of the secant over the interval from x=2 to x=2.001 is approximately 1.006001. So the answer is B. slope = 1.006001.
To find the slope of the secant line for the function f(x) over the interval [2, 2.001], we will use the slope formula:
slope = (f(2.001) - f(2)) / (2.001 - 2)
First, find the values of f(2) and f(2.001) by plugging the values of x into the given function f(x) = x^3:
f(2) = 2^3 = 8
f(2.001) = (2.001)^3 ≈ 8.006012
Now, plug these values into the slope formula:
slope = (8.006012 - 8) / (2.001 - 2) = 0.006012 / 0.001 = 6.012
The slope of the secant line over the interval is approximately 6.012. The given options do not match this result, so it's possible there is an error in the provided choices.
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You babysat your neighbor's children and they paid you $45 for 6 hours. Fill in the t-table for hours (x) and money (y)
The full t-table will be:
Hours (x) Money (y)
0 $0
1 $7.5
2 $15
3 $22.5
4 $30
5 $37.5
Given that the neighbors paid $45 for 6 hours to babysit their children.
So the rate to babysit is = $45/6 = $7.50 per hour.
So the function rule for the situation is given by,
y = 7.50x, where y is the total earning by babysitting neighbors' children and x is the number of hour to babysit.
when x = 0, y = 7.5*0 = $0
when x = 1, y = 7.5*1 = $7.5
when x =2, y = 7.5*2 = $15
when x = 3, y = 7.5*3 = $22.5
when x = 4, y = 7.5*4 = $30
when x = 5, y = 7.5*5 = $37.5
So the t-table will be:
Hours (x) Money (y)
0 $0
1 $7.5
2 $15
3 $22.5
4 $30
5 $37.5
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How can you use rational expressions and equations to describe relationship and solve problems?
Rational expressions and equations can be used to describe relationships and solve problems that involve proportions or rates of change.
They involve expressions that are fractions of polynomials, where the numerator and denominator can be thought of as values that change according to some variable.
For example, rational expressions can be used to represent rates of change in physics or finance, such as the speed of an object or the interest rate on a loan. By setting up equations with rational expressions, we can solve for unknown values or relationships between variables.
Rational equations can also be used to describe relationships between quantities that are proportional, such as the volume of a container and the amount of liquid it can hold. By solving for an unknown variable in a rational equation, we can determine the relationship between the two quantities and use it to make predictions or solve practical problems.
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Describe at least two advantages to using stem plots rather than frequency distributions
Stem plots (also known as stem-and-leaf plots) have several advantages over frequency distributions, including:
1. Stem plots show individual data points: Unlike frequency distributions, which group data into intervals, stem plots show each individual data point. This makes it easier to see the distribution of the data, including the shape, the range, and any outliers or gaps. Stem plots can also reveal patterns and trends in the data that may not be apparent from a frequency distribution.
2. Stem plots are easy to construct: Constructing a stem plot requires minimal computation and is relatively simple to create by hand. It involves only sorting the data and then splitting each data point into a stem (the leading digits) and a leaf (the trailing digit). This makes stem plots a convenient tool for quick exploratory data analysis, especially when dealing with small to moderate-sized datasets. In contrast, constructing a frequency distribution can be more time-consuming, as it requires choosing appropriate intervals and counting the number of data points in each interval.
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Rotation of 180°, followed by a dilation with scale factor 5, followed by a reflection over the line y = x.
a. A' (15, -10) b.
A' (-15, 10)
C. A' (-10, 15)
d. A' (10, -15)
Answer:
A
Step-by-step explanation:
Let's call the length of each of the other two sides x. Since the triangle is isosceles, it has two sides of equal length. Therefore, the perimeter of the triangle can be expressed as 6 + x + x Simplifying this equation, we get 2x + 6 We know that the perimeter is 22 cm so we can set up an equation and solve for x. 22 = 2x + 6 Subtracting 6 from both sides, we get 16 = 2x Dividing both sides by 2, we get x=8
Will mark brainliest two points on k are (-4, 3) and (2, -1).
write a ratio expressing the slope of k.
The ratio expressing the slope of line k is -2/3.
The ratio expressing the slope of k can be found by using the slope formula, which is: slope = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are the two given points on the line.
Plugging in the given values, we get:
slope = (-1 - 3) / (2 - (-4))
slope = -4 / 6
slope = -2/3
Therefore, the slope of the line passing through the two given points is -2/3.
To express this slope as a ratio, we can write it as:
-2:3
which means that for every decrease of 2 units in the y-coordinate, there is a corresponding decrease of 3 units in the x-coordinate.
This ratio can also be written as 2: -3 to indicate that for every increase of 2 units in the y-coordinate, there is a corresponding decrease of 3 units in the x-coordinate.
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Distribution that results in all the data intervals that have the same frequency is
called __________.
A) uniform distribution
B) bell-shaped distribution
C) skewed distribution
D)frequency distribution
Distribution that results in all the data intervals that have the same frequency is called D)frequency distribution
A frequency distribution is a way of summarizing and displaying a dataset by showing the number of times each value or range of values appears in the data.
When all the intervals in a frequency distribution have the same frequency, it means that the data is evenly distributed across those intervals. This type of distribution is useful when analyzing data that falls into discrete categories or groups, such as survey responses or test scores.
By organizing the data into intervals with equal or same frequencies, patterns in the data can become more apparent and it can be easier to draw conclusions or make predictions.
Overall, a frequency distribution is a helpful tool for understanding the distribution of data and can provide valuable insights into the characteristics of a dataset.
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A theater is selling tickets to a ''preview night'' of their new musical. The tickets cost $12 per adult and $7. 50 per child. Due to limit on seating, they can sell no more than 150 tickets. However, they would like to make at least $675 from ticket sales
it's not possible to sell at least $675 worth of tickets while also staying within the seating limit of 150 tickets. The theater may need to consider raising ticket prices or finding a larger venue to accommodate more audience members.
Let's denote the number of adult tickets sold as "A" and the number of child tickets sold as "C". Then we can set up the following system of equations to represent the given information:
A + C ≤ 150 (limit on seating)
12A + 7.5C ≥ 675 (minimum revenue required)
We want to find the possible values of A and C that satisfy these equations.
One way to solve this system is to graph the inequalities and find the region of overlap. However, since there are only two variables, we can also use substitution or elimination to solve for one variable in terms of the other.
Let's solve for A in terms of C using the first equation:
A ≤ 150 - C
Substitute this expression for A in the second equation:
12(150 - C) + 7.5C ≥ 675
Expand and simplify:
1800 - 12C + 7.5C ≥ 675
-4.5C ≥ -1125
C ≤ 250
So the number of child tickets sold must be less than or equal to 250.
Now we can substitute this inequality into the first equation to find the maximum number of adult tickets sold:
A + 250 ≤ 150
A ≤ -100
This doesn't make sense, since we can't sell negative tickets. Therefore, there is no solution that satisfies the given conditions.
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What is the interquartile range (IQR) of the data set?3,8,11,11,
12,13,15
The interquartile range (IQR) of the given data set {3, 8, 11, 11, 12, 13, 15} is 5.
How to calculate the interquartile range (IQR) for a given data set?To find the interquartile range (IQR) of a data set, follow these steps:
Order the data set in ascending order: 3, 8, 11, 11, 12, 13, 15.
Find the first quartile (Q1): This is the median of the lower half of the data set. In this case, the lower half is {3, 8, 11}. Since there is an odd number of data points, the median is the middle value, which is 8.
Find the third quartile (Q3): This is the median of the upper half of the data set. In this case, the upper half is {12, 13, 15}. The median of this set is 13.
Calculate the interquartile range (IQR): The IQR is the difference between the third quartile (Q3) and the first quartile (Q1). In this case, IQR = Q3 - Q1 = 13 - 8 = 5.
Therefore, the interquartile range (IQR) of the given data set {3, 8, 11, 11, 12, 13, 15} is 5.
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find the area of a garden that measures 6 feet by 4 feet.
Answer:
Area = Length x Width
Area = 6 feet x 4 feet
Area = 24 square feet
During a lab experiment, the
temperature of a liquid changes
from 63 °f to 102°f.
what is the percent of increase
in the temperature of the
liquid?
The percent increase in temperature of the liquid is 61.9%. This means that the temperature increased by 61.9% of its original value.
What is the percentage increase in temperature of a liquid that changes from 63°F to 102°F during a lab experiment?When we want to calculate the percent increase in a value, we need to compare the new value to the old value.
In this case, the old value is the initial temperature of the liquid, which is 63 °F, and the new value is the final temperature of the liquid, which is 102 °F.
To calculate the percent increase, we use the formula I mentioned earlier, which subtracts the old value from the new value, divides the result by the old value.
And then multiplies the quotient by 100% to express the result as a percentage.
So, for this experiment, we can calculate the percent increase in temperature as:
((102 - 63) / 63) x 100% = 61.9%
This means that the temperature of the liquid increased by 61.9% of its original value. Alternatively, we can also say that the final temperature is 161.9% of the initial temperature.
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Suppose that the weekly profit, in dollars, of producing and selling x cars is P(x) = -0.005x*-0.2x2 + 1000x - 1300. and currently 80 cars are produced and sold weekly. Use P(80) and the marginal profit when x-SAP(80)) to estimate the weekly profit of producing and selling 81 cars. Round to the nearest dollar.
• $75,731 • $75,732 • $75,645 • $77,032
To estimate the weekly profit of producing and selling 81 cars, we first need to find the current weekly profit for producing and selling 80 cars using P(80):
P(80) = -0.005(80)^(2) - 0.2(80) + 1000(80) - 1300
P(80) = $75,800
Now we need to find the marginal profit at x = 80, which is the derivative of the profit function P(x):
P'(x) = -0.01x - 0.4x + 1000
P'(80) = -0.01(80) - 0.4(80) + 1000
P'(80) = $920
The marginal profit at x = 80 is $920. This means that for each additional car produced and sold beyond 80, the profit will increase by $920.
To estimate the weekly profit of producing and selling 81 cars, we can use the approximation formula:
ΔP ≈ P'(80)Δx
where ΔP is the change in profit, Δx is the change in the number of cars produced and sold, and P'(80) is the marginal profit at x = 80.
We want to find the change in profit when the number of cars produced and sold increases from 80 to 81, so Δx = 1. Plugging in the values we have:
ΔP ≈ $920(1)
ΔP ≈ $920
This means that the estimated weekly profit for producing and selling 81 cars is:
P(81) ≈ P(80) + ΔP
P(81) ≈ $75,800 + $920
P(81) ≈ $75,720
Rounding to the nearest dollar, the answer is $75,720. So the correct option is: $75,720.
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Bonnie bought 12 bottles of pineapple juice and apple juice. The bottles of pineapple juice, p, were on sale for $1 per bottle, and the bottles of apple juice, a, were on sale for $1.75 per bottle. Bonnie spent a total of $15. How many bottles of pineapple juice and apple juice did Bonnie buy?
Answer:
Step-by-step explanation:
Let's use a system of equations to solve the problem.
We know that Bonnie bought a total of 12 bottles, so:
p + a = 12
We also know that Bonnie spent a total of $15, so:
1p + 1.75a = 15
We can solve this system of equations by substitution or elimination. Here, we'll use substitution:
p = 12 - a (from the first equation)
1(12 - a) + 1.75a = 15 (substituting p in the second equation)
12 - a + 1.75a = 15
0.75a = 3
a = 4
So Bonnie bought 4 bottles of apple juice. We can find the number of bottles of pineapple juice by substituting a=4 into the first equation:
p + 4 = 12
p = 8
Therefore, Bonnie bought 8 bottles of pineapple juice and 4 bottles of apple juice.
please answer thanksss
Answer:
this says that why is bigger than one but why is smaller than four. meaning the coordinate why could only be between one and four. and it says that x is smaller than y
so the only coordinates with whole numbers that would satisfy the inequalities would be (3,2) (2,1)
A cylindrical can, open at the top, is to hold 180 cm of liquid. Find the height and radius that minimize the amount of material needed to manufacture the can.
Let's assume that the cylindrical can has a height of "h" and a radius of "r". We want to find the values of "h" and "r" that minimize the amount of material needed to manufacture the can.
The amount of material needed to manufacture the can can be represented by the surface area of the can, which is the sum of the area of the top and bottom circles and the lateral area of the cylinder.
The area of the top and bottom circles can be calculated using the formula for the area of a circle:
A_top = A_bottom = πr^2
The lateral area of the cylinder can be calculated using the formula for the lateral surface area of a cylinder:
A_lateral = 2πrh
Therefore, the total surface area of the cylindrical can can be calculated as:
A_total = A_top + A_bottom + A_lateral
= 2πr^2 + 2πrh
Now, we need to express "h" in terms of "r" and the volume of the can, which is given as 180 cm^3. The formula for the volume of a cylinder is:
V = πr^2h
Substituting the given value of the volume and solving for "h", we get:
h = 180/(πr^2)
Substituting this expression for "h" in the equation for the total surface area, we get:
A_total = 2πr^2 + 2πr(180/(πr^2))
= 2πr^2 + 360/r
To find the values of "r" and "h" that minimize the surface area, we need to take the derivative of "A_total" with respect to "r", set it equal to zero, and solve for "r".
dA_total/dr = 4πr - 360/r^2 = 0
Solving for "r", we get:
r = (360/(4π))^(1/3) ≈ 4.35 cm
Substituting this value of "r" in the expression for "h", we get:
h = 180/(π(4.35)^2) ≈ 3.9 cm
Therefore, the height and radius that minimize the amount of material needed to manufacture the can are approximately 3.9 cm and 4.35 cm, respectively.
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The diagram represents a model of a ramp the skateboard club wants to create at the neighborhood skate park. If one pint of paint covers 200 square inches, how many pints of paint will the club need to purchase? Paint is only sold in whole pints.
The number of pints of paint that the club will need is given as follows:
2 pints of paint.
How to obtain the amount of paint?Before obtaining the amount of paint, we must obtain the surface area of the figure.
The figure in this problem is composed as follows:
One rectangle of dimensions 19 in and 8 in.One rectangle of dimensions 6 in and 19 in.Two right triangles of sides 6 in and 8 in.Hence the surface area is given as follows:
S = 19 x 8 + 19 x 6 + 2 x 0.5 x 6 x 8
S = 314 in².
If one pint of paint covers 200 square inches, hence the number of pints is given as follows:
314/200 = 2 -> rounded up, as there is not a decimal amount of pints, and one pint is not enough.
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Prove that the value of the expression: (36^5−6^9)(38^9−38^8) is divisible by 30 and 37.
_x30x37
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To prove that the expression (36^5−6^9)(38^9−38^8) is divisible by 30, we need to show that it is divisible by both 2 and 3.
First, we can factor out a 6^9 from the first term:
(36^5−6^9)(38^9−38^8) = 6^9(6^10-36^5)(38^9-38^8)
Notice that 6^10 can be written as (2*3)^10, which is clearly divisible by both 2 and 3. Also, 36 is divisible by 3, so 36^5 is divisible by 3^5. Thus, we can write:
6^9(6^10-36^5) = 6^9(2^10*3^10 - 3^5*2^10) = 6^9*2^10*(3^10 - 3^5)
Since 2^10 is divisible by 2, and 3^10 - 3^5 is clearly divisible by 3, the whole expression is divisible by both 2 and 3, and therefore divisible by 30.
To prove that the expression is divisible by 37, we can use Fermat's Little Theorem. Fermat's Little Theorem states that if p is a prime number and a is any positive integer not divisible by p, then a^(p-1) is congruent to 1 modulo p, which can be written as a^(p-1) ≡ 1 (mod p).
In this case, p = 37, and 36 is not divisible by 37. Therefore, by Fermat's Little Theorem:
36^(37-1) ≡ 1 (mod 37)
Simplifying the exponent gives:
36^36 ≡ 1 (mod 37)
Similarly, 38 is not divisible by 37, so:
38^(37-1) ≡ 1 (mod 37)
Simplifying the exponent gives:
38^36 ≡ 1 (mod 37)
Now we can use these congruences to simplify our expression:
(36^5−6^9)(38^9−38^8) ≡ (-6^9)(-1) ≡ 6^9 (mod 37)
We know that 6^9 is divisible by 3, so we can write:
6^9 = 2^9*3^9
Since 2 and 37 are relatively prime, we can use Euler's Totient Theorem to simplify 2^9 (mod 37):
2^φ(37) ≡ 2^36 ≡ 1 (mod 37)
Therefore:
2^9 ≡ 2^9*1 ≡ 2^9*2^36 ≡ 2^(9+36) ≡ 2^45 (mod 37)
Now we can simplify our expression further:
6^9 ≡ 2^45*3^9 ≡ (2^5)^9*3^9 ≡ 32^9*3^9 (mod 37)
Notice that 32 is congruent to -5 modulo 37, since 32+5 = 37. Therefore:
32^9 ≡ (-5)^9 ≡ -5^9 ≡ -1953125 ≡ 2 (mod 37)
So:
6^9 ≡ 2*3^9 ≡ 2*19683 ≡ 39366 ≡ 0 (mod 37)
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Tell whether finding the answer requires finding a greatest common factor or a least common multiple. You do not need solve the problem. A string of holiday lights at a store have three colors that flash at different times. Red lights flash every fifth second. Blue lights flash every third seconds. Green light flashes every four seconds. The store owner turns on the lights. After how many seconds will all three lights flash at the same time for the first time?
A. ) Greatest Common Factor
B. ) Lest Common Multiple
Finding the answer requires finding the least common multiple (LCM) of 5, 3, and 4, making the correct answer B. ) Least Common Multiple.
To determine whether finding the answer requires finding a greatest common factor (GCF) or a least common multiple (LCM), we need to analyze the given information.
In this scenario, the red lights flash every fifth second, the blue lights flash every third second, and the green lights flash every fourth second. We want to find the first time when all three lights flash simultaneously.
To find this time, we need to find the smallest number that is divisible by all three given numbers (5, 3, and 4). This means we are looking for the least common multiple (LCM) of these numbers.
To calculate the LCM, we can use the formula:
LCM(a, b) = (a * b) / GCF(a, b),
where GCF(a, b) represents the greatest common factor of numbers a and b.
Therefore, in this case, finding the answer requires finding the least common multiple (LCM) of 5, 3, and 4, making the correct answer:
B. ) Least Common Multiple.
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The radius of a circle is 8 centimeters. What is the area of a sector bounded by a 180° arc? Give the exact answer in simplest form.
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The area of a sector bounded by a 180° arc with a radius of 8 centimeters is 32π square centimeters in simplest form.
To find the area of a sector bounded by a 180° arc with a radius of 8 centimeters, you can follow these steps:
Step 1: Recall the formula for the area of a circle: A = πr², where A is the area and r is the radius.
Step 2: Calculate the area of the entire circle with a radius of 8 centimeters: A = π(8)² = 64π square centimeters.
Step 3: Determine the fraction of the circle represented by the 180° arc. Since a full circle is 360°, the fraction is 180°/360°, which simplifies to 1/2.
Step 4: Multiply the area of the entire circle by the fraction to find the area of the sector: (1/2) * (64π) = 32π square centimeters.
So, the area of a sector bounded by a 180° arc with a radius of 8 centimeters is 32π square centimeters in simplest form.
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Suppose the height of a cylinder is cut in half and the radius is doubled. how will this affect the volume of the cylinder ?????
Cutting the height of a cylinder in half would divide its volume by 2, while doubling the radius would multiply its volume by 4. Therefore, the overall effect on the volume of the cylinder is to multiply it by 2. So, if the original volume of the cylinder was V, after cutting the height in half and doubling the radius, the new volume would be 2V.
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