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QuestionFind and interpret (A∘r)(t)(A \circ r)(t) where r(t)=0.7tr(t)=0.7 t and A(r)=πr2A(r)=\pi r^{2}.

Studdy Solution

STEP 1

Assumptions1. The radius rr of the ripple is given by r(t)=0.7tr(t)=0.7 t . The area AA of the ripple is given by A(r)=πrA(r)=\pi r^{}
3. We are asked to find and interpret the composition of functions (A∘r)(t)(A \circ r)(t)

STEP 2

The composition of functions (A∘r)(t)(A \circ r)(t) means that we substitute the function r(t)r(t) into the function A(r)A(r).

STEP 3

Substitute r(t)r(t) into A(r)A(r).
(A∘r)(t)=A(r(t))=π(r(t))2(A \circ r)(t) = A(r(t)) = \pi (r(t))^{2}

STEP 4

Now, substitute r(t)=0.7tr(t) =0.7t into the equation.
(A∘r)(t)=π(0.7t)2(A \circ r)(t) = \pi (0.7t)^{2}

STEP 5

Calculate the square of 0.7t0.7t.
(A∘r)(t)=π(0.49t2)(A \circ r)(t) = \pi (0.49t^{2})

STEP 6

So, the function (A∘r)(t)(A \circ r)(t) is given by(A∘r)(t)=0.49πt2(A \circ r)(t) =0.49\pi t^{2}Interpretation This function describes how the area of the ripple changes over time after the pebble strikes the water. It says that the area of the ripple is proportional to the square of the time since the pebble hit the water, with the constant of proportionality being 0.49π0.49\pi square feet per second squared.

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