SOLUTION: A spherical balloon is inflated, and its diameter as a function of time is given by d(t) = t + 2 . Which of the following represents the surface area of the balloon?

Question 972689: A spherical balloon is inflated, and its diameter as a function of time is given by d(t) = t + 2 . Which of the following represents the surface area of the balloon?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the formula for surface area of a sphere is S = 4*pi * R^2
S = Surface Area of the Sphere.
pi = 3.14159........
R = radius of the sphere.

since radius of the sphere = 1/2 times the diameter, then this formula becomes:
S = 4 * pi * (D/2)^2 which becomes:
S = 4 * pi * D^2 / 4 which becomes:
S = pi * D^2.

Since you are given that D is equal to T + 2, then it stands to reason that S will be equal to pi * (T + 2)^2.

So your formula for S would be S = pi * (T + 2)^2.

let's see if this makes sense.

you have D = T + 2.

For some sample random values of T, we get:

When T = 1, D = 3
When T = 5, D = 7
When T = 10, D = 12

When D = 3, S = pi * 3^2 = pi * 9
When D = 7, S = pi * 7^2 = pi * 49
When D = 12, S = pi * 12^2 pi * 144

using the formula of S = pi * (T + 2)^2, we get:

When T = 1, S = pi * 3^2 = pi * 9
When T = 5, S = pi * 7^2 = pi * 49
When T = 10, S = pi * 12^2 = pi * 144

Let's just take one of them and see if this tracks with the formula that we started with, which was S = 4*pi*R^2

When D = 7, R = 3.5 and S = 4 * pi * (3.5)^2 = 4 * pi * 12.25 = pi * 49.
D = 7 occurs when T = 5 because D = (T + 2) becomes D = 7
The formula for S in terms of T is S = pi * (T + 2)^2 which becomes S = pi * 7^2 which becomes S = pi * 49.

I'd go with S = pi * (T + 2)^2.

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