The radius of the pond is 32 feet, under the condition that 43 feet from the bank and 75 feet from the point of tangency.
Let us consider that the center of the circle O, the point of tangency T, and Federico's position P.
We can utilize these two points to form a line. The point of tangency is the place where Federico is closest to the pond. The radius of the pond is considered perpendicular to this line and passes through the point of tangency.
Firstly, we have to the distance between Federico's position P and covers passes through points T and B (the bank). This distance is equivalent to the given radius of the circle. We have to apply the formula for the distance between a point and a line to find this distance.
Let us assume this distance as d.
d = (|BT x BP|) / |BT|
Here
|BT| = line segment length of BT,
|BP| = line segment length of BP,
BT x BP = vectors cross product of BT and BP.
Here we evaluate |BT| applying the Pythagorean theorem
|BT|² = 75²+ r²
Here,
r = radius concerning the circle.
Then,
|BP|² = 43² + r²
Staging these values into our formula for d:
d = (|BT x BP|) / |BT|
= (|BT| × |BP|) / |BT|
= |BP|
= √(43² + r²)
We want to solve for r, so we can square both sides:
d² = 43² + r²
r² = d² - 43²
r = √(d² - 43²)
Placing in d = 75,
r = √(75² - 43²)
≈ 32 feet
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The population p of a city founded in january 2009 is modeled by p(t) = 10000e*t, where t is
the time in years.
if the population was 30,000 in 2014, determine the growth rater. then, complete the model.
The population model is complete, with p(t) = 10000e(0.2197t), and the city's population is growing at a rate of about 0.2197 each year.
To determine the growth rate and complete the model for the population of a city founded in January 2009, we need to use the given information and equation, p(t) = 10000e^(rt), where t is the time in years, and r is the growth rate.
Determine the time (t) in years from January 2009 to 2014.
t = 2014 - 2009 = 5 years
Substitute the given population (30,000) and time (5 years) into the equation.
30,000 = 10000e^(5r)
Solve for the growth rate (r).
First, divide both sides by 10000:
3 = e^(5r)
Now, take the natural logarithm of both sides to isolate the exponent:
ln(3) = 5r
Finally, divide both sides by 5:
r = ln(3)/5 ≈ 0.2197
So, the growth rate is approximately 0.2197 per year.
Complete the model with the calculated growth rate.
p(t) = 10000e^(0.2197t)
The growth rate of the city's population is approximately 0.2197 per year, and the completed model for the population is p(t) = 10000e^(0.2197t).
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Gina made a
playlist of children's songs. In 1 hour,
how many more times could she play
"Row, Row, Row Your Boat" than "Twinkle,
Twinkle, Little Star"?
The number of times more that Gina can play the playlist of children's songs in an hour would be 94. 7 times.
How to find the number of times ?The playlist that Gina made of children's songs. In an hour, the number of seconds we have is :
= 60 secs x 60 mins
= 3, 600 seconds
The number of times that "Row, Row, Row Your Boat" can be played is:
= 3, 600 / 8
= 450 times
The number of times that Gina can play "Twinkle, Twinkle, Little Star" is :
= 3, 600 / 20
= 180 times
The number of times more :
= 450 - 180
= 270 times
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Joe is a college football kicker. At a point about halfway through the season he had made only 7 out of 26 field goal kicks for his team. This gives him a really lousy success rate. His coach wants his success rate to rise to 49% by Joe kicking a series of consecutive field goals successfully. How many consecutive field goals would Joe have to kick, and make, for his success rate to rise to the level his coach wants?
Joe would need to successfully kick 11 consecutive field goals to raise his success rate to 49%.
Let's use the given terms and solve the problem step by step.
1. Joe's current success rate: He made 7 out of 26 field goal kicks.
2. Desired success rate: 49%
Let's use 'x' as the number of consecutive field goals Joe needs to make to reach a 49% success rate.
Step 1: Calculate the total number of kicks after making 'x' consecutive goals.
Total kicks = 26 (previous kicks) + x (consecutive goals)
Step 2: Calculate the total number of successful kicks after making 'x' consecutive goals.
Successful kicks = 7 (previous successful kicks) + x (consecutive successful goals)
Step 3: Calculate the success rate (total successful kicks / total kicks) and set it equal to 49%.
(Successful kicks / Total kicks) = 49/100
Step 4: Substitute the expressions from Steps 1 and 2 into the equation from Step 3.
(7 + x) / (26 + x) = 49/100
Step 5: Solve for 'x'.
49 * (26 + x) = 100 * (7 + x)
1274 + 49x = 700 + 100x
49x - 100x = 700 - 1274
-51x = -574
x = 574 / 51
x ≈ 11.25
Since Joe cannot make a fraction of a goal, he needs to make 12 consecutive field goals to reach a success rate of at least 49%.
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Let {sn} be a geometric sequence that starts with an initial index of 0. the initial term is 2 and the common ratio is 5. what is s2?
The value of S2 is 50, under the condition that {sn} is a geometric sequence that starts with an initial index of 0.
Here we have to apply the principles of geometric progression.
The derived formula for regarding the nth term concerning the geometric sequence is
[tex]= ar^{n-1 }[/tex]
Here
a = first term and r is the common ratio.
For the given case from the question
a = 2
r = 5.
Then,
s2 = a× r²
= 2×5²
= 50.
A geometric sequence refers to a particular sequence of numbers that compromises each term after the first is evaluated by multiplying the previous one by a fixed one , non-zero number known as the common ratio.
For instance, if the first term of a geometric sequence is 2 and the common ratio is 5, then the sequence would be 2, 10, 50, 250.
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WILL GIVE BRAINLIEST
Tamara has decided to start saving for spending money for her first year of college. Her money is currently in a large suitcase under her bed, modeled by the function s(x) = 325. She is able to babysit to earn extra money and that function would be a(x) = 5(x − 2), where x is measured in hours. Explain to Tamara how she can create a function that combines the two and describe any simplification that can be done
To create a function that combines the two scenarios, we need to add the amount of money you earn from babysitting to the amount of money you have in your suitcase. We can represent this with the following function:
f(x) = s(x) + a(x)
Where f(x) represents the total amount of money you have after x hours of babysitting. We substitute s(x) with the given function, s(x) = 325, and a(x) with the given function, a(x) = 5(x-2):
f(x) = 325 + 5(x-2)
Simplifying this expression, we can distribute the 5 to get:
f(x) = 325 + 5x - 10
And then combine the constant terms:
f(x) = 315 + 5x
So the function that combines the two scenarios is f(x) = 315 + 5x. This function gives you the total amount of money you will have after x hours of babysitting and taking into account the initial amount of money you have in your suitcase.
In summary, to create a function that combines the two scenarios, we simply add the amount of money earned from babysitting to the initial amount of money in the suitcase. The function f(x) = 315 + 5x represents this total amount of money.
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FILL IN THE BLANK. Find the lateral (side) surface area of the cone generated by revolving the line segment y = 9/2x, 0≤ x ≤9, about the x-axis. The lateral surface area of the cone generated by revolving the line segment y 9/2x, 0≤ x ≤9 about the x-axis is _____ (Round to the nearest tenth as needed.)
The lateral surface area about x-axis is 114.1 square units.
To find the lateral surface area of the cone generated by revolving the line segment y=9/2x, 0≤x≤9 about the x-axis, we first need to find the length of the slant height of the cone.
We can think of the cone as being formed by rotating a right triangle about the x-axis.
The line segment y=9/2x intersects the x-axis at (0,0) and (9,81/2).
This forms a right triangle with base 9 and height √(81/2) = (9/2)√2.
The slant height of the cone is the hypotenuse of this right triangle, which can be found using the Pythagorean theorem:
l = √(9² + (9/2√2)²) = √(81 + 81/8) = (9/√2)√(9/8) = (9/2)√2
The lateral surface area of the cone can then be found using the formula:
L = πrl
where r is the radius of the base of the cone (which is equal to half the base of the right triangle, or 9/2) and
l is the slant height we just found.
Substituting in the values, we get:
L = π(9/2)(9/2)√2 = (81/4)π√2 ≈ 114.1
Therefore, the lateral surface area of the cone generated by revolving the line segment y=9/2x, 0≤x≤9 about the x-axis is approximately 114.1 square units.
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How many pieces of 10 5/6 inch bar can be cut from a stock 29 foot bar
20 pieces of 10 5/6 inch bar can be cut from a stock 29 foot bar.
To calculate the number of pieces of 10 5/6 inch bar that can be cut from a 29 foot bar, we need to first convert the measurements to a common unit. One foot is equal to 12 inches, so 29 feet equals 348 inches.
Next, we need to determine how many 10 5/6 inch bars can be cut from the 348-inch stock bar. To do this, we can use division. First, we need to convert the mixed number 10 5/6 to an improper fraction by multiplying the whole number by the denominator and adding the numerator. This gives us 125/6 inches.
Now, we can divide the length of the stock bar (348 inches) by the length of one 10 5/6 inch bar (125/6 inches). This gives us:
348 / (125/6) = 20.736
Since we cannot cut a partial bar, we need to round down to the nearest whole number. Therefore, we can cut 20 pieces of 10 5/6 inch bar from a 29 foot stock bar.
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5. copy the table and find the quantities marked *. (take t = 3)
curved
total
surface
area
area
*
2
2
vertical surface
object radius height
(a) cylinder
4 cm
72 cm
*
(b) sphere
192 cm2
(c) cone
4 cm
60 cm?
*
(d) sphere
0.48 m²
(e) cylinder
5 cm
(f) cone 6 cm
(g) cylinder
* * *
330 cm?
225 cm
108 m2
2
2 m
The table shows the calculated curved surface area, total surface area, and vertical surface area for various geometric objects, including cylinders, cones, and spheres. The missing values are found for each object, with a given value of t = 3.
Radius is 4 cm
Height is 72 cm
curved surface area of cylinder
2πrt = 2π(4)(72) = 576π cm²
total surface area
2πr(r+h) = 2π(4)(76) = 304π cm²
vertical surface area
2πrh = 2π(4)(72) = 576π cm²
Radius is 4 cm
Height is 60 cm
curved surface area of cylinder of cone
πr√(r²+h²) = π(4)√(4²+60²) = 124π cm²
total surface area
πr(r+√(r²+h²)) = π(4)(4+√(4²+60²)) = 140π cm²
vertical surface area
πr√(r²+h²) = π(4)√(4²+60²) = 124π cm²
total surface area of sphere
0.48 m² = 48000 cm²
curved surface area of cylinder
Radius is 5 cm
Height 2 m = 200 cm
2πrt = 2π(5)(200) = 2000π cm²
total surface area
2πr(r+h) = 2π(5)(205) = 2050π cm²
vertical surface area
2πrh = 2π(5)(200) = 2000π cm²
curved surface area of cylinder
Radius is 6 cm
Height 10 cm
πr√(r²+h²) = π(6)√(6²+10²) = 34π cm²
total surface area
πr(r+√(r²+h²)) = π(6)(6+√(6²+10²)) = 78π cm²
vertical surface area
πr√(r²+h²) = π(6)√(6²+10²) = 34π cm²
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If the ratio of ambers miniature house to the original structure is 2:35 and the miniature requires 4 square feet of flooring how much flooring exists in the original house
The original house has 70 square feet of flooring.
If the ratio of the miniature house to the original structure is 2:35, then we can say that the miniature house is 2/35th the size of the original house in terms of floor area. Let's assume that the original house has x square feet of flooring. Then, we can set up a proportion based on the ratios:
2/35 = 4/x
Solving for x, we get:
x = 70
Therefore if the ratio of ambers miniature house to the original structure is 2:35 and the miniature requires 4 square feet of the flooring then original house has 70 square feet of flooring.
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For exercise, a softball player ran around the bases 12 times in 15 minutes. At the same rate, how many times could the bases be circled in 50 minutes?
The bases could be circled 40 times in 50 minutes at the same rate.
To solve this problemFor this issue's solution, let's use unit rates.
In order to calculate the unit rate,
Considering that the player went 12 times around the bases in 15 minutes, the unit rate is 12/15, = 0.8 times per minute.
In a minute, the player would have circled the bases 0.8 times. By dividing the unit rate by the number of minutes, we can calculate how many times the bases could be circled in 50 minutes:
50 minutes x 0.8 times each minute = 40 times.
Therefore, the bases could be circled 40 times in 50 minutes at the same rate.
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What was the average amount of books read per student according to the histogram below?
The average amount of books read per student according to the histogram is given as follows:
1.27 books.
How to calculate the mean of a data-set?The mean of a data-set is given by the sum of all observations in the data-set divided by the number of observations, which is also called the cardinality of the data-set.
The histogram shows the number of times for each observation, hence:
7 students read zero books.9 students read one book.6 students read two books.4 students read three books.Hence the mean is calculated as follows:
M = (7 x 0 + 9 x 1 + 6 x 2 + 4 x 3)/(7 + 9 + 6 + 4) = 1.27 books.
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The spinner at the right is spun 12 times. it lands on blue 1 time.
1. what is the experimental probability of landing on blue?
2. compare the experimental and theoretical probabilities of the spinner landing on blue. if the probabilities are not close, explain a possible reason for the discrepancy.
Experimental probability of landing on blue = 1/12 and experimental probability and theoretical probability are not close.
1.
To find the experimental probability of landing on blue, we need to divide the number of times it landed on blue by the total number of spins.
Experimental probability of landing on blue = Number of times landed on blue / Total number of spins
Here, the spinner was spun 12 times and landed on blue 1 time.
Experimental probability of landing on blue = 1/12
2.
The theoretical probability of landing on blue is the ratio of the number of blue spaces to the total number of spaces on the spinner. Since there is only one blue space out of four total spaces, the theoretical probability is 1/4 or 0.25.
The experimental probability = 1/12 = 0.083
So, the experimental probability and theoretical probability are not close.
A possible reason for the discrepancy is likely due to the small sample size of spins. With a larger number of spins, the experimental probability should converge closer to the theoretical probability. This is known as the law of large numbers in probability theory.
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Which is the better deal: an account that pays 4% interest compounded daily or one that pays 3.95% compounded continuously?
Answer:
compounded continuously
Step-by-step explanation:
compounded continuously occurs more frequently than daily
In ΔGHI, h = 9. 6 cm, g = 9. 3 cm and ∠G=109°. Find all possible values of ∠H, to the nearest 10th of a degree
The two possible values for angle H in triangle GHI are approximately 93.1 degrees and 273.1 degrees, rounded to the nearest tenth of a degree
How to find possible angle in GHI triangle?To find the possible values of angle H in triangle GHI, we can use the law of cosines.
Let's label angle H as x. Then, we can use the law of cosines to solve for x:
cos(x) = (9.3² + 9.6² - 2(9.3)(9.6)cos(109))/ (2 * 9.3 * 9.6)
Simplifying this equation, we get:
cos(x) = -0.0588
To solve for x, we can take the inverse cosine of both sides:
x = cos⁻ ¹ (-0.0588)
Using a calculator, we can find that x is approximately 93.1 degrees.
However, there is another possible value for angle H. Since cosine is negative in the second and third quadrants,
We can add 180 degrees to our previous result to find the second possible value for angle H:
x = 93.1 + 180 = 273.1 degrees
So the two possible values for angle H are approximately 93.1 degrees and 273.1 degrees, rounded to the nearest tenth of a degree.
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What are measurements less than 435 inches???? Hurry it’s due tomorrow!!!
Any measurement below 435 inches qualifies as a value less than 435 inches.
To find measurements less than 435 inches, you simply need to consider any value below 435 inches. Here's a step-by-step explanation:
1. Understand the question: You are looking for measurements less than 435 inches.
2. Identify the range: The range includes all values below 435 inches.
3. Provide examples: Examples of measurements less than 435 inches can be 400 inches, 350 inches, 250 inches, 100 inches, and so on.
Remember, any measurement below 435 inches qualifies as a value less than 435 inches.
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The largest single rough diamond ever found, the cullinan diamond, weighed 3106 carats; how much does the diamond weigh in miligrams? in pounds? (1 carat - 0. 2 grams)
the diamond weighs mg.
the diamond weighs lbs
If the largest single rough diamond ever found, the Cullinan diamond, weighed 3106 carat, it weighs approximately 621,200 milligrams and 1.37 pounds.
The Cullinan Diamond, the largest single rough diamond ever found, weighed 3,106 carats. To convert its weight to milligrams and pounds, we'll use the conversion factor of 1 carat = 0.2 grams.
First, convert carats to grams:
3,106 carats * 0.2 grams/carat = 621.2 grams
Next, convert grams to milligrams:
621.2 grams * 1,000 milligrams/gram = 621,200 milligrams
Lastly, convert grams to pounds:
621.2 grams * 0.00220462 pounds/gram ≈ 1.37 pounds
So, the Cullinan Diamond weighs approximately 621,200 milligrams and 1.37 pounds.
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FILL IN THE BLANK. The function f(x) = 4x³ – 12x² – 576x + 6 = is decreasing on the interval (______ , ______ ). It is increasing on the interval (-[infinity], _____ ) and the interval (_____ , [infinity]). The function has a local maximum at _______
The function has a local maximum at x = -6.
To determine the intervals on which the function f(x) = 4x³ - 12x² - 576x + 6 is increasing or decreasing, we first find its derivative, f'(x), and then analyze its critical points.
f'(x) = 12x² - 24x - 576
Now, set f'(x) = 0 and solve for x:
12x² - 24x - 576 = 0
Divide by 12:
x² - 2x - 48 = 0
Factor:
(x - 8)(x + 6) = 0
So, the critical points are x = 8 and x = -6.
Analyze the intervals:
f'(-7) > 0, so increasing on (-∞, -6)
f'(0) < 0, so decreasing on (-6, 8)
f'(9) > 0, so increasing on (8, ∞)
The function f(x) is decreasing on the interval (-6, 8). It is increasing on the interval (-∞, -6) and the interval (8, ∞). The function has a local maximum at x = -6.
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Leroy is building a slide for his kids. If the ladder is 5 feet tall and he wants the bottom of the slide to be 12 feet from the ladder, how long does the slide need to be?
We can use the Pythagorean theorem to solve this problem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Let x be the length of the slide. Then we have a right triangle with legs of length 5 (the height of the ladder) and x, and hypotenuse of length 12 (the distance from the ladder to the bottom of the slide).
Using the Pythagorean theorem:
12^2 = 5^2 + x^2
144 = 25 + x^2
Subtracting 25 from both sides:
119 = x^2
Taking the square root of both sides:
x ≈ 10.91
Therefore, the slide needs to be about 10.91 feet long.
Given the following demand function, q = D(x) = 1536 - 2x², find the following: a. The elasticity function, E(x). b. The elasticity at x = 20. c. At x = 20, demand (circle one) is elastic has unit elasticity is inelastic d. Find the value(s) of x for which total revenue is a maximum (assume x is in dollars).
a. The elasticity function: E(x) = -8x²/(1536-2x²)
b. The elasticity at x = 20 is -2.78.
c. At x = 20, demand is elastic.
d. The value of x for which total revenue is a maximum is $12.
a. The elasticity function, E(x), can be calculated using the formula:
E(x) = (dQ/Q) / (dx/x)
where Q is the quantity demanded and x is the price. In this case, we have:
Q = D(x) = 1536 - 2x²
Taking the derivative with respect to x, we get:
dQ/dx = -4x
Using this, we can calculate the elasticity function:
E(x) = (dQ/Q) / (dx/x) = (-4x/(1536-2x²)) * (x/Q) = -8x²/(1536-2x²)
b. To find the elasticity at x = 20, we substitute x = 20 into the elasticity function:
E(20) = -8(20)²/(1536-2(20)²) = -3200/1152 = -2.78
So the elasticity at x = 20 is -2.78.
c. To determine whether demand is elastic, unit elastic, or inelastic at x = 20, we can use the following guidelines:
If E(x) > 1, demand is elastic.
If E(x) = 1, demand is unit elastic.
If E(x) < 1, demand is inelastic.
Since E(20) = -2.78, demand is elastic at x = 20.
d. To find the value(s) of x for which total revenue is a maximum, we use the formula for total revenue:
R(x) = xQ(x) = x(1536 - 2x²)
Taking the derivative of R(x) with respect to x, we get:
dR/dx = 1536 - 4x²
Setting this equal to zero to find the critical points, we get:
1536 - 4x² = 0
Solving for x, we get:
x = ±12
To determine whether these are maximum or minimum points, we take the second derivative of R(x):
d²R/dx² = -8x
At x = 12, we have d²R/dx² < 0, so R(x) is maximized at x = 12. Therefore, the value of x for which total revenue is a maximum is $12.
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If 3 quarts is greater then 4 prints is that an equivalent measure
If 3 quarts is greater than 4prints, then the measure is not equivalent.
What is equivalent measurement?Equivalent units can be used to convert different units to the same unit for comparison. Equivalent means equal. For example , 1 kilogram is equal to 1,000 grams.
For example,
3 teaspoons = 1 tablespoon.
4 tablespoons = 1/4 cup.
5 tablespoons + 1 teaspoon = 1/3 cup.
8 tablespoons = 1/2 cup.
1 quart = 2pints
therefore 3 quarts = 2×3 = 6pints
therefore the statement that 3 quarter is greater than 4 prints is true and not an equivalent measure.
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QUESTION IN PHOTO I MARK BRAINLIEST
The value of x in the intersecting chord is determined as 18.6.
What is the value of x?The value of x is calculated by applying intersecting chord theorem, which states that the angle at center is equal to the arc angle of the two intersecting chords.
m ∠EDF = arc angle EF
50 = 5x - 43
The value of x is calculated as follows;
5x = 50 + 43
5x = 93
divide both sides by 5;
5x/5 = 93/5
x = 18.6
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A local deli sells 6-inch sub sandwiches for $2.95. Now the deli has decided to sell a “family sub” that is 50 inches long. If they want to make the larger sub price comparable to the price of the smaller sub, how much should it charge? Show all work.
Deli should charge $24.50 for the 50-inch family sub.
How much should the deli charge for a 50-inch?In a transaction, the price of something refers to amount of money that you have to pay in order to buy it. To make the prices comparable, we can use the unit price which is as follows>
The price per inch of 6-inch sub is:
= $2.95 / 6 inches
= $0.49/inch
To make 50-inch sub price, we will solve as:
= $0.49/inch * 50 inches
= $24.50
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CAN SOMEONE HELP PLEASE!
A restaurant is serving a special lunch combo meal that includes a drink, a main dish, and a dessert. Customers can choose from 5 drinks, 6 main dishes, and 3 desserts.
How many different combo meals are possible?
Select from the drop-down menu to correctly complete the statement.
Customers can create (14, 39, 60, 120) different lunch combo meals.
8 Real / Modelling An advertising company uses a graph of this
equation to work out the cost of making an advert:
y=10+0.5x
where x is the number of words and y is the total cost of the bill in
pounds.
a)Where does the line intercept the y-axis?
b)How much is the bill when there are no words in the advert?
c)What is the gradient of the line?
d)How much does each word cost?
The gradient in the given equation is 0.5.
The given linear equation is y=10+0.5x where x is the number of words and y is the total cost of the bill in pounds.
a) When x=0, we get y=10
So, at (0, 10) the line intercept the y-axis.
b) $10 is the bill when there are no words in the advert.
c) Compare y=0.5x+10 with y=mx+c, we get m=0.5
So, the gradient of the line is 0.5
d) From equation, we can see the cost of each word is $0.5.
Therefore, the gradient in the given equation is 0.5.
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Evaluate the following integral using u-substituion: indefinite integral dx/|x|*sqrt4x^2-16
The solution to the integral is ∫ dx/|x|*√4x²-16 is ∫ dx/|x|*√(4x²-16) = 2 ln|sin(θ)| + CC
How to explain the integralWe can then rewrite the integral in terms of u as:
∫ dx/|x|*√(4x²-16) = ∫ du/|u|*√(u²-16)
Next, we can use another substitution of the form u = 4sec(θ), which will transform the integrand into: 2/(|sec(θ)|*√(sec²(θ)-1)) dθ
Using the identity sec²(θ)-1=tan²(θ), we can simplify the integrand to:
2/(|sec(θ)|sqrt(sec²(θ)-1)) = 2/(|sec(θ)||tan(θ)|)
We can then split the integral into two parts, corresponding to the two possible signs of sec(θ):
∫ du/|u|*√(u²-16) = 2 ∫ dθ/(sec(θ)tan(θ))
= 2 [ ∫ dθ/(sec(θ)tan(θ)), for sec(θ)>0
∫ dθ/(-sec(θ)tan(θ)), for sec(θ)<0 ]
The integral ∫ dθ/(sec(θ)tan(θ)) can be solved using the substitution u = sin(θ), which gives:
∫ dθ/(sec(θ)tan(θ)) = ∫ du/u = ln|u| + C = ln|sin(θ)| + C
Therefore, the indefinite integral is:
∫ dx/|x|*√(4x²-16) = 2 ln|sin(θ)| + C
where θ satisfies the equation 4sec(θ) = 2x.
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Given the objective Function: Revenue = 75x+85y and the critical points: (0,0) (180,120) (300,0)
A large apartment complex has 1,500 units, which are filling up at a rate of 10% per month. If the
apartment complex starts with 15 occupied units, what logistic function represents the number of
units occupied over time?
ON(t)
1500
1+114e-0. 101
ON(t)
800
1+114e-0. 101
N(t)
800
1+99e-0. 100
N(t)
1500
1+99e-0. 101
The logistic function that represents the number of units occupied over time is given by:
[tex]N(t) = (K / (1 + A * e^(-r*t))),[/tex]
where N(t) is the number of units occupied at time t, K is the carrying capacity (maximum number of units that can be occupied),
A is the initial amount of units occupied, r is the growth rate, and e is the base of the natural logarithm.
In this case, the carrying capacity K is 1500 units, and the initial amount of occupied units A is 15 units. The growth rate r can be calculated as follows:
[tex]r = ln((10%)/(100% - 10%)) = ln(0.1/0.9) ≈ -0.101[/tex]
Substituting the given values into the logistic function, we get:
[tex]N(t) = (1500 / (1 + 15 * e^(-0.101*t)))[/tex]
Simplifying further, we get:
[tex]N(t) = (100 / (1 + e^(-0.101*t))) + 15[/tex]
Therefore, the logistic function that represents the number of units occupied over time is:
[tex]N(t) = (100 / (1 + e^(-0.101*t))) + 15[/tex], where t is measured in months.
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what is 7 + 9d = 7d +3?
Answer:
-2
Step-by-step explanation:
7+9d=7d+3
7+2d=3
2d=-4
d=-2
The perimeter of an isosceles triangle is 51 in. One side is 18 in and another is 15 in. What is the length of the missing side?
The length of the missing side is equal to 18 inches.
How to calculate the perimeter of this triangle?In Mathematics and Geometry, the perimeter of a triangle can be calculated by using this mathematical equation:
P = a + b + c
Where:
P represents the perimeter of a triangle.a, b, and c represents the side lengths of a triangle.By substituting the given parameters or dimensions into the formula for the perimeter of a triangle, we have the following;
51 = 18 + 15 + x
51 = 33 + x
x = 51 - 33
x = 18 inches.
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Find the volume generated when the area bounded by the curve y?=x, the line x=4 and the
x-axis is revolved about the y-axis.
To find the volume generated, we need to use the formula for volume of revolution. We are revolving the area bounded by the curve y=x, the line x=4 and the x-axis about the y-axis.
First, we need to find the limits of integration for x. The curve y=x intersects the line x=4 at y=4, so we integrate from x=0 to x=4.
Next, we need to find the radius of the rotation. The radius is the distance from the y-axis to the curve at each value of x. Since we are revolving about the y-axis, the radius is simply x.
Using the formula for volume of revolution, we get:
V = π∫(radius)^2 dx from 0 to 4
V = π∫x^2 dx from 0 to 4
V = π[x^3/3] from 0 to 4
V = π[(4^3/3) - (0^3/3)]
V = (64π/3)
Therefore, the volume generated when the area bounded by the curve y=x, the line x=4 and the x-axis is revolved about the y-axis is (64π/3).
To find the volume generated when the area bounded by the curve y=x^2, the line x=4, and the x-axis is revolved around the y-axis, we'll use the disk method. The formula for the disk method is:
Volume = π * ∫ [R(x)]^2 dx
Here, R(x) is the radius function and the integral is taken over the given interval on the x-axis. In this case, R(x) = x and the interval is from 0 to 4.
Volume = π * ∫ [x]^2 dx, with the integral from 0 to 4
Now, we'll evaluate the integral:
Volume = π * [ (1/3)x^3 ](0 to 4)
Volume = π * [ (1/3)(4)^3 - (1/3)(0)^3 ]
Volume = π * [ (1/3)(64) - 0 ]
Volume = π * [ (64/3) ]
So, the volume generated is (64/3)π cubic units.
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