Answer:
The composite function is:
a(r(t)) = π (0.5 + 2t)²
After simplification it becomes:
a(r(t)) = π (0.25 + 2t + 4t²)
Step-by-step explanation:
Here is the complete function:
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.
Solution:
The composite function is written by applying one function to another. The requirement of this question is to write a composite function to find are of region at time t.
Let
a(r) = πr²
is function 1
and
r(t) = 0.5 + 2t
is function 2
Then
This means that function r(t) should be applied to a(r) and it should be inside function a(r)
Then composite function becomes:
a(r(t))
Now put the value of r(t) = 0.5 + 2t into a(r) = πr²
a(r(t)) = π (0.5 + 2t)²
So this is the composite function to find area of the region at time t
Simplifying this equation we get:
a(r(t)) = π (0.5 + 2t)²
= π (0.5 + 2t) (0.5 + 2t)
= π [(0.5 * 0.5)+ (0.5 * 2t) + (2t * 0.5) + (2t *2t)]
= π (0.25 + 1t + 1t + 4t²)
= π ( 0.25 + t + t + 4t²)
= π (0.25 + 2t + 4t²)
a(r(t)) = π (0.25 + 2t + 4t²)
