Question

The radius of a circular puddle is growing at a rate of 20 cm/sec. (a) how fast is its area growing at the instant when the radius is 40 centimeters? hint [see example 1.] (round your answer to the nearest integer.) incorrect: your answer is incorrect. cm2/s (b) how fast is the area growing at the instant when it equals 64 square centimeters? hint [use the area formula to determine the radius at that instant.] (round your answer to the nearest integer.)

Answer 1

Part (a)
 The expression for the circle area is given by:
 A = pi * r ^ 2
 Where,
 r: radio
 We now derive the expression of the area with respect to time:
 A '= 2 * pi * r * r'
 Substituting values:
 A '= 2 * pi * (40) * (20)
 A '= 5026.6 cm ^ 2 / s
 Answer:
 its area is growing at the instant when the radius is 40 centimeters at:
 A '= 5026.6 cm ^ 2 / s
 Part b)
 A = pi * r ^ 2
 We look for the radio:
 r = root (A / pi)
 r = root (64 / pi)
 r = 4.5 cm
 We now derive the expression of the area with respect to time:
 A '= 2 * pi * r * r'
 Substituting values:
 A '= 2 * pi * (4.5) * (20)
 A '= 565.5 cm ^ 2 / s
 Answer:
 its area is growing at:
 A '= 565.5 cm ^ 2 / s

Answer 2

(a) The rate at which area grow at a radius of 40cm will be

$\dfrac{dA}{dt} =5026.6\ \dfrac{cm^2}{sec}$

(b) The rate at which area grow when the area is 64 $cm^2$ will be

$\dfrac{dA}{dt} =565.5\ \dfrac{cm^2}{sec}$

What will be the rate of growth of the area?

Here we have

rate of radius  $\dfrac{dr}{dt} = 20\ \dfrac{cm}{sec}$

(a) Area growth at radius 40cm

Area will be $A=\pi r^2$

Differentiating the area w.r.t. time

$\dfrac{dA}{dt} =2\pi r \dfrac{dr}{dt}$

By putting the values

$\dfrac{dA}{dt} =2\pi\times 40\times 20$

$\dfrac{dA}{dt} =5026.6 \dfrac{cm^2}{sec}$

(b) The area growth when the area is 64 $cm^2$

Again from the area formula

$A=\pi r^2$

$r= \sqrt{\dfrac{A}{\pi} }$

$r=\sqrt{\dfrac{64}{\pi} }$

$r=4.5\ cm$

Now the rate of the area

$\dfrac{dA}{dt} =2\pi r\dfrac{dr}{dt}$

$\dfrac{dA}{dt} =2\pi \times 4.5\times 20$

$\dfrac{dA}{dt} =565.5\ \dfrac{cm^2}{sec}$

Thus

(a) The rate at which area grow at a radius of 40cm will be

$\dfrac{dA}{dt} =5026.6\ \dfrac{cm^2}{sec}$

(b) The rate at which area grow when the area is 64 $cm^2$ will be

$\dfrac{dA}{dt} =565.5\ \dfrac{cm^2}{sec}$

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