Answers
Check the picture below.
so let's recall that the area of a circle is just πr² where r = radius, so since the sizes of each pizza are 10, 14, 18 and 22, that's simply their diameter, so that means they each have a radius half of that, or namely 5, 7, 9 and 11, as you see in the picture, so, hmmm which radius gives us the most area per the fraction provided, namely per slices provided?
[tex]\cfrac{4}{8}\pi 5^2\implies \cfrac{25\pi }{2} ~~ \approx ~~ 39.26~in^2 \\\\[-0.35em] ~\dotfill\\\\ \cfrac{3}{8}\pi 7^2\implies \cfrac{147\pi }{8} ~~ \approx ~~ 57.73~in^2 \\\\[-0.35em] ~\dotfill\\\\ \cfrac{2}{8}\pi 9^2\implies \cfrac{81\pi }{4}~~ \approx ~~ 63.62~in^2 ~~ \textit{\LARGE \checkmark} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{1}{8}\pi 11^2\implies \cfrac{121\pi }{8}~~ \approx ~~ 47.52~in^2[/tex]
Answer: Large
Step-by-step explanation: Hope it helps Good luck!!!
Related Questions
Give the rectangular coordinates for the point
Answers
The rectangular coordinates from the polar coordinates are: (-4√2, -4√2)
How to convert polar coordinates to rectangular coordinates?The steps to to convert polar coordinates to rectangular coordinates are:
Step 1: Find the x -coordinate for the rectangular coordinate form of the point by using the equation x = r cos(θ)
Step 2: Find the y -coordinate for the rectangular coordinate form of the point by using the equation y = rsin(θ)
Step 3: Write the rectangular coordinates as (x,y) using the results from steps 1 and 2.
We are given the polar coordinates as (8, 225°)
Thus:
x-coordinate of rectangular coordinate = 8 cos 225 = -4√2
y-coordinate of rectangular coordinate = 8 sin 225 = -4√2
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Brooklyn and Matthew both recently got a job and want to start a savings account to earn interest on money they save. Brooklyn's paycheck for was for $625 net pay-after taxes. the bank her parents use offer savings account compounded monthly with an interest rate of 2.5%. Matthews paycheck was also for $446 net pay. his parents know of a savings account that earns 4% compounded quarterly.
a. use the compounding interest rate formula to identify who will have more money after two years (assuming no additional deposits or withdrawals are made)
b. use the Desmos graphing calculator to graph Brooklyn's equation and Matthew’s equation on the same graph. identify the points of intersection and interpret the results.
WHAT MISTAKES DID THE STUDENTS MAKE??
Answers
After two years, Brooklyn will have more money in her savings account than Matthew.
Brooklyn's savings account grows at a slower rate than Matthew's initially but eventually catches up and surpasses Matthew's account due to the higher interest rate and monthly compounding.
a.
[tex]A = P(1 + r/n)^{nt}[/tex]
where A is the final amount, P is the initial principal (or net pay), r is the interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the number of years.
For Brooklyn saving account,
P = $625, r = 0.025, n = 12 (monthly compounding), and t = 2.
A = $625(1 + 0.025/12)^(12*2) = $686.71
For Matthew saving account,
P = $446, r = 0.04, n = 4 (quarterly compounding), and t = 2.
A = $446(1 + 0.04/4)^(4*2) = $506.84
b.
To graph Brooklyn's equation and Matthew's equation on the same graph, we can use the following equations:
Brooklyn's equation: A = 625(1 + 0.025/12)^(12t)
Matthew's equation: A = 446(1 + 0.04/4)^(4t)
Now,
The resulting graph will show the growth of each saving account over time.
The points of intersection on the graph represent the times when the two savings accounts have the same amount of money.
From the graph, we can see that the two accounts intersect at around 11 months and 23 months.
We can say that Brooklyn's savings account grows at a slower rate than Matthew's initially, but eventually catches up and surpasses Matthew's account due to the higher interest rate and monthly compounding.
Thus,
After two years, Brooklyn will have more money in her savings account than Matthew.
Brooklyn's savings account grows at a slower rate than Matthew's initially but eventually catches up and surpasses Matthew's account due to the higher interest rate and monthly compounding.
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A geography teacher assigns each student to write a report about one of the first 13 colonies. Students select the name of a colony by "blindly" drawing a colony's name from a bag. Once a colony has been drawn, it is not replaced.
What is the probability that the first student selects Pennsylvania and the second student selects Virginia? Round your answer to the nearest hundredth of a percent.
Answers
The probability that the first student selects Pennsylvania and the second student selects Virginia to the nearest hundredth of a percent is 0.64%.
Probability problemThe probability that the first student selects Pennsylvania is 1/13. Once Pennsylvania has been drawn, there are only 12 colonies left in the bag, so the probability that the second student selects Virginia is 1/12.
To find the probability that both events occur, we multiply the probabilities:
(1/13) x (1/12) = 1/156
To express this probability as a percentage, we multiply by 100:
(1/156) x 100 ≈ 0.64%
Therefore, the probability that the first student selects Pennsylvania and the second student selects Virginia is approximately 0.64%.
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Keng adds a 3-inch-wide frame around all sides of his canvas.
Answers
Answer:
That statement describes an action taken by Keng to add a 3-inch-wide frame around all sides of his canvas, likely for artistic or aesthetic purposes. This would result in the canvas being extended by 3 inches in each direction, effectively increasing its overall dimensions and adding a border or frame around the original artwork. The purpose and effect of adding a frame may vary depending on the artistic intention of the artist, but it is a common practice in art and design to enhance the presentation and visual appeal of the artwork.
Step-by-step explanation:
Question 1 (1 point)
Write an inequality for the sentence.
The stadium held less than 25,000.
B
O a
O b
9ba580e611107c96c9efb866417dc160.webm 64 KB
s> 25,000
$≤25,000
Oc $<25,000
Answers
The inequality that represents the sentence "The stadium held less than 25,000 people" is given as follows:
c < 25,000.
What are the inequality symbols?The four inequality symbols, along with their meaning on the number line and the coordinate plane, are presented as follows:
> x: the amount is greater than x -> the number is to the right of x with an open dot at the number line. -> points above the dashed horizontal line y = x on the coordinate plane.< x: the amount is less than x. -> the number is to the left of x with an open dot at the number line. -> points below the dashed horizontal line y = x on the coordinate plane.≥ x: the amount is at least x. -> the number is to the right of x with a closed dot at the number line. -> points above the solid vertical line y = x on the coordinate plane.≤ the amount is at most x. -> the number is to the left of x with a closed dot at the number line. -> points above the dashed vertical line y = x on the coordinate plane.The amount is less than in this problem, hence the symbol is given as follows:
<.
As the amount is less than 25000, the inequality is given as follows:
c < 25,000.
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Which equation represents a direct variation?
Oy - 4x = 8
Oy + 2 = 7x
Oy - 3x = 0
O y = 5x - 2
Answers
The only equation that represents a direct variation is: y - 3x = 0
How to identify direct variation?Direct Variation is defined as the relationship that exists between two variables in which one is a constant multiple of the other. For example, when one variable changes the other, then they are said to be in proportion. If b is directly proportional to a the equation is of the form b = ka (where k is a constant).
Making y the subject in each of the options gives:
A) y = 4x + 8
B) y = 7x - 2
C) y = 3x
D) y = 5x - 2
Looking at them, the only one where there is a relationship that exists between two variables in which one is a constant multiple of the other is option C
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A group consists of seven Democrats and eight Republicans. Four people are selected to attend a conference.
a. In how many ways can four people be selected from this group of fifteen?
b. In how many ways can four Republicans be selected from the eight Republicans?
c. Find the probability that the selected group will consist of all Republicans.
a. The number of ways to select four people from the group of fifteen is
b. The number of ways to select four Republicans from the group of eight Republicans is
c. The probability is
Answers
There 1365 ways to choose four people from the group of fifteen.
b. There are 70 ways to choose four Republicans from the group of eight Republicans.
C. The probability is about 0.0513, or 5.13%.
What is the probability about?a. To know the ways that four people can be selected from this group of fifteen is by:
nCr = n! / (r! x (n-r)!),
Where:
n = total number of items
r = is the number of items to be selected,
! = the factorial of a number.
Putting in the values into the the formula:
15C4 = 15! / (4! x (15-4)!)
(15-4)! = 11!
15C4 = 1365
B. Since:
n = 8
r = 4
Putting in the values into the the formula:
8C4 = 8! / (4! x (8-4)!)
(8-4)! = 4!
8C4 = 70
c. The Probability = Number of ways to choose four Republicans / Number of ways to choose four people
Hence Probability = 70 / 1365
= 0.0513
Therefore, the probability that the selected group will consist of all Republicans is about 0.0513, or 5.13%.
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construct a binomial whose gcf is 7a^3
Answers
The binomial whose greatest common factor is 7a³ is 14a⁴ + 35a³.
Given, the a³ is not common in A and C.
Now, in 14a⁴ + 35a³
14a⁴ = 2 x 7 x a³ x a
35a³ = 7 x 5 x a³
So, the common factors of 14 and 35 is 7.
Then, 14a⁴ + 35a³
= ( 2 x 7 x a³ x a) + ( 5 x 7 x a³)
= 7a³ (2a + 5)
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geometry pls help fast 13 and 14
Answers
Answer:
13: (2,3) 14: A. Yes, 90, 180, 270 B. No C. No
Step-by-step explanation:
13. Right: -2+4, Down: 5-2
14. If it can be rotated and be similar, it has rotational.
Find the angle between the pair of vectors to the nearest tenth of the degree
Answers
The value of angle between the two vectors is 86⁰.
What is the angle between the two vectors?The value of angle between the two vectors is calculated as follows;
tan θ = vy/vx
where;
vy is the sum of the vertical directionvx is the sum of vectors in horizontal direction( -8, 9), (-9, -6)
vy = (-6 - 9) = -15
vx = (-9 + 8) = -1
tan θ = ( -15 ) / ( -1 )
tan θ = 15
The value of θ is calculated by taking arc tan of the fraction,;
θ = tan ⁻¹ ( 15 )
θ = 86⁰
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Determine if the statement is true or false.
Angle AEC and DEB are complementary angles and therefore add up to 180 degrees.
1. True
2. False
Answers
The statement "Angle ∠AEC and ∠DEB are complementary angles" is false.
Supplementary angle - Two angles are said to be supplementary angles if their sum is 180 degrees.
Complementary angle - Two angles are said to be complementary angles if their sum is 90 degrees.
Angle ∠AEC and ∠DEB are supplementary angles because the sum is 180 degrees.
Thus, the given statement is false.
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The following blueprint of a kitchen has dimensions of 7 inches by 7 inches. The island has been highlighted in red.
The island's actual dimensions are 3 1/2 feet by 1 3/4 feet. If the scale of the blueprint is 1 inch = 2 feet, what are the dimensions of the island on the blueprint?
Answers
The dimensions of the island on the blueprint are 14 inches by 3.5 inches.
We have,
The actual dimensions of the island are 3 1/2 feet by 1 3/4 feet.
We need to find the dimensions of the island on the blueprint, given that the scale of the blueprint is 1 inch = 2 feet.
To convert the actual dimensions to the dimensions on the blueprint, we need to use the scale factor of 1 inch = 2 feet.
We can set up a proportion to relate the actual dimensions to the dimensions on the blueprint:
Actual dimension/blueprint dimension = scale factor
Let x be the length of the island on the blueprint.
Then we can set up the following proportion:
3.5 feet / (1.75 feet)
= x inches / 7 inches
Simplifying,
2 = x / 7
Multiplying both sides by 7, we get:
x = 14 inches
The length of the island on the blueprint is 14 inches.
Similarly, we can find the width of the island on the blueprint:
1.75 feet / 3.5 feet
= y inches / 7 inches
Simplifying, we have:
0.5 = y / 7
Multiplying both sides by 7, we get:
y = 3.5 inches
The width of the island on the blueprint is 3.5 inches.
Thus,
The dimensions of the island on the blueprint are 14 inches by 3.5 inches.
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A curve, described by x2 + y2 + 6y = 0, has a point A at (−3, −3) on the curve.
Part A: What are the polar coordinates of A? Give an exact answer.
Part B: What is the polar form of the equation? What type of polar curve is this?
Part C: What is the directed distance when theta equals 4 pi over 3 question mark Give an exact answer.
Answers
a) The polar coordinates of point A are (√(18), π/4).
b) The curve is a circle centered at the origin with radius 6.
c) The directed distance is the value of r, which is 6 √(3).
To find the polar coordinates of point A on the curve, we need to convert the point from Cartesian to polar coordinates. The conversion formula is:
r = √(x² + y²)
θ = arctan(y/x)
Using the values of point A, we have:
r = √((-3)² + (-3)²) = √(18)
θ = arctan((-3)/(-3)) = arctan(1) = π/4
To find the polar form of the equation x² + y² + 6y = 0, we need to convert it from Cartesian to polar coordinates. The conversion formulae are:
x = r cos(θ)
y = r sin(θ)
Using these formulae, we can rewrite the equation as:
r² cos²(θ) + r² sin²(θ) + 6r sin(θ) = 0
Simplifying this equation, we get:
r = -6 sin(θ) / (1 - cos²(θ))
To find the directed distance when θ equals 4 π over 3, we need to substitute this value of θ into the polar equation we found in Part B. Doing so, we get:
r = -6 sin(4 π/3) / (1 - cos²(4 π/3))
r = -6(-√(3)/2) / (1 - (-1/2)²)
r = 6 √(3)
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Find the value of the following expression and round to the nearest integer:
Answers
The value of the Expression is 610, 919.
We have,
Expression : [tex]\sum_{n=0}^{61[/tex] 700 (1.07[tex])^{n+1[/tex]
The expression is a summation formula, representing the sum of a series of values.
Here r= 1.07 and n is number of terms.
So, [tex]\sum_{n=0}^{61[/tex] 700 (1.07[tex])^{n+1[/tex]
= 700 (1) + [tex]\sum_{n=0}^{61[/tex] 700 (1.07[tex])^{n[/tex]
= 700 + 700 ([tex]r^{61[/tex] - 1)/ (r-1)
= 700 + 700 ([tex](1.07)^{61[/tex]-1)/ (0.07)
= 700+ 700 (871.7428)
= 610, 919
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Solve for m. m/2 - squareroot 5 = 3
Answers
Answer:
We can start by isolating the variable term on one side of the equation and moving all other terms to the other side.
First, we'll add the square root of 5 to both sides of the equation:
m/2 - sqrt(5) + sqrt(5) = 3 + sqrt(5)
Simplifying:
m/2 = 3 + sqrt(5)
Next, we'll multiply both sides of the equation by 2:
m = 2(3 + sqrt(5))
Simplifying:
m = 6 + 2sqrt(5)
Therefore, the value of m is 6 + 2sqrt(5).
Step-by-step explanation:
Forty people were asked their favorite kind of pizza. Thirty percent of the people surveyed chose sausage. How many people preferred sausage?
Answers
So, 30% of 40 is:
0.30 x 40 = 12
Therefore, 12 people preferred sausage.
Answer: 12 people
Step-by-step explanation:
0.30 x 40 = 12
Margarita fue a la tienda y compro una cartera y unos jeans por un
total de RD$3,250. Sabiendo que las cartera excede al jeans en
RD$970, ¿Cuántos pago margarita por cada artículo?
Cartera = RDS
Jeans = RD$
Escribir las respuestas numéricas y sin comas.
OK
Answers
The solution is , price of jeans = RD$ 1140 and, price of handbag =
RD$ 2110.
Here, we have,
given that,
Margarita went to the store and bought a bag and some jeans for a total of RD$3,250.
Knowing that the handbag exceeds the jeans by RD$970,
now, we have to find that, how many do she pay for each article.
let, price of jeans = RD$ x
so, price of handbag = RD$ (x +970)
ATQ, we get,
RD$ x + RD$ (x +970) = RD$3,250
or, RD$ 2x + 970 = RD$3,250
or, RD$ 2x = RD$ 2280
or, x = RD$ 1140
Hence, price of jeans = RD$ 1140 and, price of handbag = RD$ 2110.
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A belt runs a pulley at 80 revolutions per minute. Find the angular velocity of the pulley in radians per second.
Answers
A belt runs a pulley at 80 revolutions per minute.
(a) To find the angular speed of the pulley in radians per second, we can use the formula
ω = 2πf
Where
ω = angular speed in radians per second
f = frequency in Hertz
We know that the pulley rotates at a rate of 80 revolutions per minute. To convert this to a frequency in Hertz, we can divide by 60 seconds per minute
f = 80 rev/min ÷ 60 min/s = 4/3 Hz
Now we can plug in this value for f into the formula
ω = 2πf = 2π(4/3) = 8π/3 rad/sec
Therefore, the angular speed of the pulley is 8π/3 radians per second.
(b) To find the linear speed of the belt in centimeters per second, we can use the formula
v = rω
Where
v = linear speed in centimeters per second
r = radius of the pulley in centimeters
ω = angular speed in radians per second
We know the radius of the pulley is given in centimeters. Let's assume it is r cm. We just found the angular speed to be 8π/3 radians per second. Now we can plug in these values into the formula
v = rω = r(8π/3) = (8π/3)r cm/s
Therefore, the linear speed of the belt is (8π/3)r centimeters per second.
The given question is incomplete and the complete question is ''A belt runs a pulley at 80 revolutions per minute. Find the angular velocity of the pulley in radians per second. (b)Find the linear speed of the belt in centimeters per second''.
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Alice finds shirts on sale for $18.99.She buys twelve how much money does she spend?
Answers
Answer:
Well, if Alice buys 12 shirts which each cost $18.99 the equation would be 18.99 * 12 which = 227.88
Alice spent $227.88 on 12 shirts
Step-by-step explanation:
Answer:
Well, if Alice buys 12 shirts which each cost $18.99 the equation would be 18.99 * 12 which = 227.88
Step-by-step explanation:
The distance between cities A and B on a map is 12.5 in. The distance from city B to city C, is 8.5 in, and the distance from C to A is 16.25 in. If the bearing
from A to B is N75°E, find the bearing from C to 4. Round to the nearest tenth of a degree.
16.25 in
12.5 in
8.5 in
The bearing from city C to city 4 is approximately (Choose one) (Choose one)
Answers
The bearing of C to A is 239⁰.
What is the bearing of C to A?
The bearing of C to A is calculated by finding the value of angle C using cosine rule since we know the value of all the sides of the triangle.
AB² = AC² + CB² - 2(AC)(CB) cosC
12.5² = 16.25² + 8.5² - 2(16.25 x 8.5) x cos C
156.25 = 336.3125 - 276.25cosC
276.25cosC = 180.06
cosC = 180.06/276.25
cos (C) = 0.6518
C = cos⁻(0.6518)
C = 49.3⁰
The value of angle A is calculated as follows;
Sin A/CB = Sin C/AB
sin A/8.5 = sin 49.3/12.5
sin A = 8.5 [sin 49.3/12.5]
sin A = 0.5157
A = sin⁻¹ (0.5157)
A = 31⁰
The bearing of C to A is calculated as;
= 270⁰ - 31⁰
= 239⁰
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You can use the notation P(A), read “the probability of event B, given event A” to write a
A. Probability distribution
B. Frequency table
C. Conditional probability
D. Cumulative probability
Answers
You can use the notation P(A), read “the probability of event B, given event A” to write a conditional probability. The correct answer is C.
Conditional probability refers to the probability of one event occurring given that another event has already occurred. In this case, we are interested in the probability of event B occurring given that event A has already occurred, and we can represent this using the notation P(B|A), where '|' means 'given'.
For example, let's say we are interested in the probability of getting a head on a coin toss (event B), given that the coin was flipped and landed on heads (event A). We could represent this using the notation P(B|A). The value of P(B|A) would be 1, because if the coin already landed on heads, then the probability of getting a head on the next flip is certain.
Conditional probability is an important concept in probability theory and is often used in real-world applications, such as predicting the likelihood of a disease given certain symptoms, or the probability of an event occurring given certain conditions.
The correct answer is C.
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Help with this page :)
Answers
17. 15 degrees
18. ABC
19. 5.88
20. 11.59
let me know if you'd like an explanation
Need help. This please
Answers
The domain of the quadratic function in this problem is given as follows:
All real values.
How to obtain the domain of the function?The domain of a function is the set of all the possible input values that can be assumed by the function.
On the graph, the domain of the function is given by the values of x of the function.
A quadratic function has no restrictions on the domain, hence it is defined by all the real values.
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Please help asap!!!!! I'm confused
Answers
The area of the parallelogram is 8.5 square miles. This is found by multiplying the length of one of the parallel sides, 2 miles, by the height, which is given as 4 1/4 miles.
To find the area of a parallelogram, we can multiply the length of one of its parallel sides by the length of its perpendicular height. Therefore, to find the area of this parallelogram, we need to determine its height.
We are given that one of the parallel sides has a length of 4 1/4 mi and the other has a length of 2 mi. We are also given that the length of the perpendicular on one of the parallel sides is 4 1/4 mi, which means that this is the height of the parallelogram.
So, the area of the parallelogram is
Area = base x height
Area = 2 mi x 4 1/4 mi
Area = 8 1/2 mi²
Therefore, the area of the parallelogram is 8 1/2 square miles or 8.5 square miles.
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Is this right. Don’t know the difference between the stratified and systematic random. Pic below
Answers
The difference between the stratified and systematic random sampling is explained below.
Systematic random sampling is a method where every Kth person of the population is chosen to be part of the sample, whereas stratified random sampling is a method where the population is first divided into subgroups, and then drawing a simple random sample from each subgroup.
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Helppp I need the ending numbers ?
Answers
The solution is, the simplification of the equation is :
4x^2 - 4 = (2x -2 ) (2x+2)
Here, we have,
given that,
the expression is:
4x^2 - 4
now, we have to simplify this.
we know that ,
The a^2 - b^2 formula is also known as "the difference of squares formula". The a square minus b square is used to find the difference between the two squares without actually calculating the squares.
It is one of the algebraic identities.
It is used to factorize the binomials of squares.
The a^2 - b^2 formula is given as:
a^2 - b^2 = (a - b) (a + b).
so, we have,
4x^2 - 4
= (2x)^2 - 2^2
= (2x -2 ) (2x+2)
Hence, The solution is, the simplification of the equation is :
4x^2 - 4 = (2x -2 ) (2x+2)
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What is the value of z?
Answers
Answer: z = 8
Step-by-step explanation:
The diagram shows us that 8z + 3z + 2 = 90
so we can say that: 11z = 88
therefore z = 8.
Note: diagrams can be misleading! this diagram technically shows us that 64 < 26!
how do you find the net of a rectangular prism
Answers
Answer:
The formula looks like this:
Surface Area = 2 l w + 2 l h + 2 h w ,where SA = surface area, l = length, w = width, and h = height. In the rectangular prism net above, l = 8 inches, w = 5 inches, and h = 3 inches. Simply put these numbers into the formula and solve for surface area.
math hw for tonight
help solve this problem! Thank you!
ap cal bc
Answers
Answer:
first option
Step-by-step explanation:
differentiate using the power rule
[tex]\frac{d}{dx}[/tex] (a[tex]x^{n}[/tex] ) = na[tex]x^{n-1}[/tex]
then
[tex]\frac{dy}{dx}[/tex] = [tex]\frac{dy}{dt}[/tex] × [tex]\frac{dt}{dx}[/tex] = [tex]\frac{\frac{dy}{dt} }{\frac{dx}{dt} }[/tex]
y = t² + 4t
[tex]\frac{dy}{dt}[/tex] = 2t + 4
x = t - 3
[tex]\frac{dx}{dt}[/tex] = 1
then
[tex]\frac{dy}{dx}[/tex] = [tex]\frac{2t+4}{1}[/tex] = 2t + 4
Answer:
2t + 4
Step-by-step explanation:
A parametric equation is one where x and y are defined separately in terms of a third variable (often the parameter t).
To find dy/dx from parametric equations, differentiate each equation with respect to the parameter t, then use the chain rule:
[tex]\boxed{\dfrac{\text{d}y}{\text{d}x}=\dfrac{\text{d}y}{\text{d}t} \times \dfrac{\text{d}t}{\text{d}x}}[/tex]
Differentiate the two parametric equations with respect to t:
[tex]x=t-3 \implies \dfrac{\text{d}x}{\text{d}t}=1[/tex]
[tex]y=t^2+4t \implies \dfrac{\text{d}y}{\text{d}t}=2t+4[/tex]
Use the chain rule to combine them:
[tex]\begin{aligned}\implies \dfrac{\text{d}y}{\text{d}x}&=\dfrac{\text{d}y}{\text{d}t} \times \dfrac{\text{d}t}{\text{d}x}\\\\&=(2t+4) \times \dfrac{1}{1}\\\\&=2t+4\end{aligned}[/tex]
Therefore:
[tex]\boxed{\dfrac{\text{d}y}{\text{d}x}=2t+4}[/tex]
An insurance company offers an ordinary annuity that earns 6.5% interest compounded annually. A couple plans to make equal annual deposits into this account for 30 years and then make 20 equal annual withdrawals of €25,000, reducing the balance of the account to zero.
(i) Compute the value of the fund based on the withdrawals required. [5 marks]
(ii) Compute the amount of each deposit needed in order to maintain the fund. [5 marks]
(iii) Compute the total interest earned over the entire 50 years. [5 marks]
Answers
Answer:
(i) To compute the value of the fund based on the withdrawals required, we can use the formula for the future value of an annuity due:
FV = P * ((1 + r)^n - 1) / r) * (1 + r)
where FV is the future value of the annuity, P is the annual payment, r is the interest rate per period, n is the total number of periods, and the extra (1 + r) factor is because the payments are made at the beginning of each period.
In this case, P = €25,000, r = 0.065, n = 20. We want to find the future value at the end of the 20-year period:
FV = 25000 * ((1 + 0.065)^20 - 1) / 0.065) * (1 + 0.065)
FV ≈ €743,704.96
Therefore, the value of the fund based on the withdrawals required is approximately €743,704.96.
(ii) To compute the amount of each deposit needed in order to maintain the fund, we can use the formula for the present value of an ordinary annuity:
PV = P * ((1 - (1 + r)^(-n)) / r)
where PV is the present value of the annuity, P is the annual payment, r is the interest rate per period, and n is the total number of periods.
In this case, PV = €743,704.96, r = 0.065, n = 20. We want to find the annual payment:
PV = P * ((1 - (1 + 0.065)^(-20)) / 0.065)
P ≈ €22,630.53
Therefore, the amount of each deposit needed in order to maintain the fund is approximately €22,630.53.
(iii) To compute the total interest earned over the entire 50 years, we can subtract the total deposits from the total withdrawals, and then subtract the initial balance. The total deposits are the annual deposit amount times the number of years (30), and the total withdrawals are the annual withdrawal amount times the number of years (20). The initial balance is the present value of the annuity that we calculated in part (ii).
Total deposits = €22,630.53 * 30 = €678,915.90
Total withdrawals = €25,000 * 20 = €500,000
Initial balance = €743,704.96
Total interest earned = Total withdrawals - Total deposits - Initial balance
Total interest earned = €500,000 - €678,915.90 - €743,704.96
Total interest earned ≈ -€922,620.86
Note that the negative sign indicates that the insurance company actually earned interest on this annuity, rather than the couple earning interest on their investment. This is because the withdrawals are greater than the deposits, and the interest rate earned by the insurance company is greater than the interest rate paid to the couple.
Step-by-step explanation: