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A circular oil spill continues to increase in size. The radius of the oil spill, in miles, is given by the function [tex]r(t) = 0.5 + 2t[/tex], where [tex]t[/tex] is the time in hours. The area of the circular region is given by the function [tex]A(r) = \pi r^2[/tex], where [tex]r[/tex] is the radius of the circle at time [tex]t[/tex].

Explain how to write a composite function to find the area of the region at time [tex]t[/tex].



Answer :

GinnyAnswer
Certainly! To find the area of the circular oil spill at any given time [tex]\( t \)[/tex], we need to create a composite function. This composite function will combine the radius of the oil spill as a function of time with the area of the circular region as a function of the radius.

Let's break it down step-by-step.

### Step 1: Radius as a Function of Time
Given the radius of the oil spill in miles as a function of time [tex]\( t \)[/tex] in hours:
[tex]\[ r(t) = 0.5 + 2t \][/tex]

### Step 2: Area as a Function of Radius
Given the area of the circular region in square miles as a function of the radius [tex]\( r \)[/tex] in miles:
[tex]\[ A(r) = \pi r^2 \][/tex]

### Step 3: Composite Function for the Area as a Function of Time
To find the area of the oil spill at time [tex]\( t \)[/tex], we need to substitute the radius function [tex]\( r(t) \)[/tex] into the area function [tex]\( A(r) \)[/tex]. This will give us a composite function [tex]\( A(r(t)) \)[/tex] representing the area as a function of time.

1. Substitute the radius function into the area function:
[tex]\[ r(t) = 0.5 + 2t \][/tex]
[tex]\[ A(r) = \pi r^2 \][/tex]

2. Express [tex]\( A \)[/tex] in terms of [tex]\( t \)[/tex]:
[tex]\[ A(r(t)) = A(0.5 + 2t) \][/tex]

3. Compute the area function:
Since [tex]\( A(r) = \pi r^2 \)[/tex], we replace [tex]\( r \)[/tex] with [tex]\( 0.5 + 2t \)[/tex]:
[tex]\[ A(0.5 + 2t) = \pi (0.5 + 2t)^2 \][/tex]

4. Simplify the expression:
[tex]\[ (0.5 + 2t)^2 = (0.5 + 2t)(0.5 + 2t) \][/tex]
Using the distributive property:
[tex]\[ (0.5 + 2t)^2 = 0.5^2 + 2 \cdot 0.5 \cdot 2t + (2t)^2 \][/tex]
[tex]\[ = 0.25 + 2t + 4t^2 \][/tex]

Now multiply by [tex]\( \pi \)[/tex] to get the area:
[tex]\[ A(0.5 + 2t) = \pi (0.25 + 2t + 4t^2) \][/tex]

Therefore:
[tex]\[ A(t) = \pi (0.25 + 2t + 4t^2) \][/tex]

### Conclusion
The composite function that gives the area of the circular oil spill at time [tex]\( t \)[/tex] is:
[tex]\[ A(t) = \pi (0.25 + 2t + 4t^2) \][/tex]

This function allows us to find the area of the oil spill at any given time [tex]\( t \)[/tex].

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