A pebble is dropped into a calm pond,causing ripples in the form of concentric circles.The radius (in feet)of the outer ripple is $r ( t ) = 0.2 t$ ,where t is the time in seconds after the pebble strikes the water.The area of the circle is given by the function $A ( r ) = \pi r ^ { 2 }$ .Find and interpret $( A \circ r ) ( t )$ .
A) $$( A \circ r ) ( t ) = 0.2 \pi t$$ ; $$( A \circ r ) ( t )$$ represents the area of the circle at time t.
B) $( A \circ r ) ( t ) = 0.2 \pi t ^ { 2 }$ ; $( A \circ r ) ( t )$ represents the area of the circle at time t.
C) $$( A \circ r ) ( t ) = 0.04 \pi t$$ ; $( A \circ r ) ( t )$ represents the area of the circle at time t.
D) $( A \circ r ) ( t ) = 0.04 \pi t ^ { 2 }$ ; $$( A \circ r ) ( t )$$ represents the area of the circle at time t.
E) $$( A \circ r ) ( t ) = 0.04 \pi t ^ { 3 }$$ ; $( A \circ r ) ( t )$ represents the area of the circle at time t.

Question

A pebble is dropped into a calm pond,causing ripples in the form of concentric circles.The radius (in feet)of the outer ripple is $r ( t ) = 0.2 t$ ,where t is the time in seconds after the pebble strikes the water.The area of the circle is given by the function $A ( r ) = \pi r ^ { 2 }$ .Find and interpret $( A \circ r ) ( t )$ . ​
A) $$( A \circ r ) ( t ) = 0.2 \pi t$$ ; $$( A \circ r ) ( t )$$ represents the area of the circle at time t.
B) $( A \circ r ) ( t ) = 0.2 \pi t ^ { 2 }$ ; $( A \circ r ) ( t )$ represents the area of the circle at time t.
C) $$( A \circ r ) ( t ) = 0.04 \pi t$$ ; $( A \circ r ) ( t )$ represents the area of the circle at time t.
D) $( A \circ r ) ( t ) = 0.04 \pi t ^ { 2 }$ ; $$( A \circ r ) ( t )$$ represents the area of the circle at time t.
E) $$( A \circ r ) ( t ) = 0.04 \pi t ^ { 3 }$$ ; $( A \circ r ) ( t )$ represents the area of the circle at time t.

Quizplus · Accepted Answer

$( A \circ r ) ( t ) = A(r(t)) = A(0.2t) = \pi (0.2t)^2 = 0.04\pi t^2$, which represents the area of the outer ripple at time $t$.

A Pebble Is Dropped into a Calm Pond,causing Ripples in the Form