Question

A circle's radius that has an initial radius of 0 cm is increasing at a constant rate of 5 cm per second.

a. Write a formula to expresses the radius of the circle, r (in cm), in terms of the number of seconds, t since the circle started growing.r=
b. Write a formula to express the area of the circle, A (in square cm), in terms of the circle's radius, r (in cm). A =
c. Write a formula to expresses the circle's area, A (in square cm), in terms of the number of seconds, t, since the circle started growing. A =
d. Write your answer to part (c) in expanded form - so that your answer does not contain parentheses. A =

Answer 1

a)
`r(t)=5t`

b)
`A=\pi\cdot r^2`

c)
`A=\pi\cdot (5t)^2`

d)
`A=25\pi t^2`

Explanation:

We know that the circle is increasing its radio from an initial state of r=0 cm, at a rate of 5 cm/s.

This can be expressed as:

`r(0)=0\\\\dr/dt=5\\\\r(t)=r(0)+dr/dt\cdot t=0+5t\\\\r(t)=5t`

a) Radius of the circle, r (in cm), in terms of the number of seconds, t since the circle started growing:

`r(t)=5t`

b) Area of the circle, A (in square cm), in terms of the circle's radius, r (in cm):

`A=\pi\cdot r^2`

c) Circle's area, A (in square cm), in terms of the number of seconds, t, since the circle started growing:

`A=\pi\cdot r^2\\\\A=\pi\cdot (5t)^2`

d) Expanded form for the area A:

`A=\pi\cdot (5t)^2=25\pi\cdot t^2`