Composing Functions Quick Check

Question

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Question
A car is leaking oil. The oil that has leaked spreads in a circular layer. The area of the circular layer of oil is given by A(r)=πr2
. The radius of the circular layer of oil is increasing every hour, which is given by the function r(t)=0.5t3
. What function can be used to determine the area of the circular layer of oil after t hours? What is the domain?(1 point)
Responses

(A∘r)(t)=0.25πt6
and the domain is t≥0
.
open paren cap A circ r close paren times t is equal to 0 point 2 5 pi t to the 6th power and the domain is t is greater than or equal to 0 .

(A∘r)(t)=0.5πt6
and the domain is (−∞, ∞)
.
open paren cap A circ r close paren times t is equal to 0 point 5 pi t to the 6th power and the domain is open paren negative infinity comma infinity close paren .

(A∘r)(t)=0.5π3t6
and the domain is t≥0
.
open paren cap A circ r close paren times t is equal to 0 point 5 pi cubed t to the 6th power and the domain is t is greater than or equal to 0 .

(A∘r)(t)=0.5π3t6
and the domain is (−∞, ∞)
.
open paren cap A circ r close paren times t is equal to 0 point 5 pi cubed t to the 6th power and the domain is open paren negative infinity comma infinity close paren .

Answer 1

To determine the area of the circular layer of oil after \( t \) hours, we need to compose the two functions given:

1. The area of a circle, given by \( A(r) = \pi r^2 \).
2. The radius function, given by \( r(t) = 0.5t^3 \).

The composition of the functions is given by \( (A \circ r)(t) = A(r(t)) \).

First, substitute \( r(t) \) into the area function:

\[ A(r(t)) = A(0.5t^3) = \pi (0.5t^3)^2 \]

Calculating this, we have:

\[ (0.5t^3)^2 = 0.25t^6 \]

Thus, the area function becomes:

\[ A(r(t)) = \pi \cdot 0.25t^6 = 0.25\pi t^6 \]

So the composite function is:

\[ (A \circ r)(t) = 0.25\pi t^6 \]

Now, for the domain of this function, since \( t \) represents time and cannot be negative, the domain is \( t \geq 0 \).

Therefore, the correct response is:

\[ (A \circ r)(t)=0.25\pi t^6 \text{ and the domain is } t \geq 0. \]