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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):
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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