MATH SOLVE

3 months ago

Q:
# The radius of a circular puddle is growing at a rate of 25 cm/s.(a) How fast is its area growing at the instant when the radius is 50 cm? HINT [See Example 1.] (Round your answer to the nearest integer.)(b) How fast is the area growing at the instant when it equals 36 cm2? HINT [Use the area formula to determine the radius at that instant.] (Round your answer to the nearest integer.)

Accepted Solution

A:

Answer:a) 2500π cm/sb) 300√π cm/sStep-by-step explanation:Given:Rate of growth of radius, [tex]\frac{dr}{dt}[/tex] = 25 cm/sArea of circle is given as:A = πr²a)Rate of growth of area, [tex]\frac{dA}{dt}=\frac{d(\pi r^2)}{dt}[/tex]or⇒ [tex]\frac{dA}{dt}=(2)\pi r\frac{dr}{dt}[/tex] ............(1)at r = 50 cmon substituting the respective values, we get
⇒ [tex]\frac{dA}{dt}=(2)\pi r\frac{dr}{dt}[/tex]or⇒ [tex]\frac{dA}{dt}[/tex] = 2π(50)25 = 2500π cm/sb) when area , A = 36 cm²36 = πr²r = [tex]\frac{6}{\sqrt{\pi}}[/tex]thus, using (1)⇒ [tex]\frac{dA}{dt}=(2)\pi r\frac{dr}{dt}[/tex]on substituting the respective values, we get
⇒ [tex]\frac{dA}{dt}=(2)\pi (\frac{6}{\sqrt{\pi}})25[/tex]or⇒ [tex]\frac{dA}{dt}[/tex] = 300√π cm/s