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Antonio drops a glass marble into a pond and creates ripples that form concentric circles on the surface of the water. The radius of the circle, [tex]r[/tex], in centimeters is given by the function [tex]r(t)=1+4t[/tex], where [tex]t[/tex] is the time in seconds that the ripples move outward from the center of the circle. Find the function [tex]A(t)[/tex] that represents the area, [tex]A[/tex], of the expanding circle in [tex]t[/tex] seconds.

A. [tex]A(t)=\pi(16t^2+8t+1)[/tex]

B. [tex]A(t)=\pi(16t^2+4t+1)[/tex]

C. [tex]A(t)=16\pi t^2+1[/tex]

D. [tex]A(t)=4\pi t^2+1[/tex]



Answer :

GinnyAnswer
To solve for the function [tex]\( A(t) \)[/tex] that represents the area of the expanding circle in [tex]\( t \)[/tex] seconds, we need to follow these steps:

1. Identify the radius function [tex]\( r(t) \)[/tex]:
Given: [tex]\( r(t) = 1 + 4t \)[/tex]

2. Recall the formula for the area of a circle:
The area [tex]\( A \)[/tex] of a circle is given by [tex]\( A = \pi r^2 \)[/tex].

3. Substitute the radius function [tex]\( r(t) \)[/tex] into the area formula:
[tex]\( A(t) = \pi [r(t)]^2 \)[/tex]
Hence, [tex]\( A(t) = \pi (1 + 4t)^2 \)[/tex]

4. Expand the squared term [tex]\( (1 + 4t)^2 \)[/tex]:
[tex]\[ (1 + 4t)^2 = (1 + 4t)(1 + 4t) \][/tex]
Using the distributive property or binomial expansion:
[tex]\[ (1 + 4t)^2 = 1^2 + 2(1)(4t) + (4t)^2 = 1 + 8t + 16t^2 \][/tex]

5. Substitute the expanded form back into the area formula:
[tex]\[ A(t) = \pi (1 + 8t + 16t^2) \][/tex]

6. Distribute π across each term:
[tex]\[ A(t) = \pi + 8\pi t + 16\pi t^2 \][/tex]

So, the function [tex]\( A(t) \)[/tex] representing the area of the expanding circle in [tex]\( t \)[/tex] seconds is:
[tex]\[ A(t) = \pi + 8\pi t + 16\pi t^2 \][/tex]

Now, we match this result with the given options:
[tex]\[ A. \ A(t) = \pi\left(16 t^2 + 8 t + 1\right) \][/tex]
[tex]\[ B. \ A(t) = \pi\left(16 t^2 + 4 t + 1\right) \][/tex]
[tex]\[ C. \ A(t) = 16 \pi t^2 + 1 \][/tex]
[tex]\[ D. \ A(t) = 4 \pi t^2 + 1 \][/tex]

We can see that option A matches our derived function:

[tex]\[ A(t) = \pi (1 + 8t + 16t^2) = \pi\left(16 t^2 + 8 t + 1\right) \][/tex]

Therefore, the correct answer is:
A. [tex]\( A(t) = \pi\left(16 t^2 + 8 t + 1\right) \)[/tex]

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