A circular oil spill continues to increase in size. The radius of the oil spill, in miles, is given by the function r(t) = 0.5 + 2t, where t is the time in hours. The area of the circular region is given by the function A(r) = πr2, where r is the radius of the circle at time t.
Explain how to write a composite function to find the area of the region at time t.
A(t) = π (r(t))^2 = π (0.5+2t)^2
To find the area of the circular region at time t, we need to use a composite function that combines the functions r(t) and A(r).
Step 1: Start with the function r(t) = 0.5 + 2t, which gives us the radius of the oil spill at time t.
Step 2: Substitute this expression for r(t) into the function A(r) = πr^2, where r is the radius of the circle.
So, we replace r in A(r) with the function r(t).
A(r(t)) = π(r(t))^2
Step 3: Simplify the composite function.
A(r(t)) = π(0.5 + 2t)^2
This composite function, A(r(t)) = π(0.5 + 2t)^2, will give us the area of the circular region at time t.
To find the area of the circular region at a given time t, we need to use a composite function that combines the two given functions: r(t) = 0.5 + 2t and A(r) = πr^2.
1. Start with the function r(t) = 0.5 + 2t, which gives us the radius of the oil spill at time t.
2. Substitute the expression for r(t) into the function A(r) = πr^2. This will allow us to find the area of the region at time t.
3. The composite function A(t) combines the functions r(t) and A(r).
4. Substitute the expression for r(t) into the function A(r), which gives us A(t) = π(0.5 + 2t)^2.
5. Simplify the expression A(t) = π(0.5 + 2t)^2 to get the formula for the area of the circular region at a given time t.
Therefore, the composite function to find the area of the region at time t is A(t) = π(0.5 + 2t)^2.