SOLUTION: The area of two circles are directly proportional to their radii squared (A/A=r^2/r^2) Find the area if A=78.5, r=5 and r=8

Question 1074931: The area of two circles are directly proportional to their radii squared (A/A=r^2/r^2) Find the area if A=78.5, r=5 and r=8
Found 2 solutions by josgarithmetic, Theo:
Answer by josgarithmetic(39620) (Show Source): You can put this solution on YOUR website!
This cannot be:
r=5 and r=8

but this was not what you meant.

What you are trying to find is like this:

OR

Depending on if the unknown circle area is for the smaller or the larger circle.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the areas of 2 circles are directly proportional to their radii squared.

let A equal the area of the first circle.
let x equal the radius of the first circle.

let B equal the area of the second circle.
let y equal the radius of the second circle.

the area of the first circle is equal to pi * x^2
the area of the second circle is equal to pi * y^2

therefore you get:

A = pi * x^2
B = pi * y^2

if you divide A by B, then you get:

A/B = (pi * x^2) / (pi * y^2)

pi in the numerator and denominator cancel out and you are left with:

A/B = x^2 / y^2

you know x and you know y and you know A.

x is the radius of the first circle.
A is the area of the first circle.
y is the radius of the second circle.

the formula becomes:

78.5/B = 5^2/8^2

solve for B to get B = (78.5 * 8^2) / 5^2.

this makes B equal to 200.96

that's close, but no cigars because the area of A was rounded to start with.

to confirm, simply calculate the area of each circle based on their radius.

A = pi * 5^2 = 25 * pi = 78.539816234

B = pi * 8^2 = 64 * pi = 201.0619298

if you took pi * 8^2 and divided it by pi * 5^2, you would get a ratio of 8^2 / 5^2 = 2.56.

if you multiplied 78.539816234 * 2.56, you would get 201.0619298.

in other words, the areas of the 2 circles are in direct proportion to the square of their radii.

your solution would be 78.5 * 8^2 / 5^2 = 200.96 as the area of the circle with a radius of 8.

if you were to solve this using the direct variation formula of y = k * x, you would do the following.

y = area of the circle.

x = radius of the circle squared = r^2.

k is the constant of variation.

when y = 78.5 and x = 5^2, the formula becomes:

78.5 = k * 5^2

solve for k to get k = 58.5 / 5^2 = 3.14

if k looks like it's the value of pi, it's because it is.

now you have solved for k and you have a radius of 8.

your formula becomes y = 3.14 * 8^2 = 200.96.

you get the same answer of 200.96.

it appears the authors of the problem were using the value of 3.14 for pi rather than the more accurate value of 3.141592654.

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