Question 1074931
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.