Question

Band members form a circle of radius r when the music starts. They march outward as they play. The function f(t) = 2.5t gives the radius of the circle in feet after t seconds. Using g(r) = ar^2 for the area of the circle, identify the composite function that gives the area of the circle after t seconds.

A) g(f(t)) = 6.25t^2
B) og(f(1)) = 12.5t
C) of(g()) = 6.25912
D) og(f(t)) = 6.25t^3

Answer 1

The composite function that gives the area of the circle after t seconds is g(f(t)) = 6.25πt^2, which corresponds to option A provided the constant 'a' in the function g(r) = ar^2 is the area constant π.

Step-by-step explanation:

The student is being asked to identify the composite function that represents the area of the circle after t seconds, given the radius as a function of time f(t) = 2.5t and the area as a function of radius g(r) = ar^2. To find this, we must substitute f(t) into g(r), obtaining g(f(t)), which is a(2.5t)^2. Assuming that a, the constant in g(r), represents the area constant π (since the area of a circle is πr^2), the function simplifies to g(f(t)) = π(2.5t)^2. As this constant is not provided, we assume it to be π and find that the composite function that correctly represents the area of the circle after t seconds is g(f(t)) = 6.25πt^2, which corresponds to option A with a being π.