A mold culture in a dorm refrigerator is circular and growing. The radius is increasing at a rate of 0.1 cm/day. How fast is the area growing when the culture is 6 centimeters in radius (The area of a disc of radius r is $A = \pi r ^ { 2 }$ .)
A) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 18 \pi \frac { \mathrm { cm } } { \mathrm { day } }$
B) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 0.6 \pi \frac { \mathrm { cm } } { \mathrm { day } }$
C) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 2.4 \pi \frac { \mathrm { cm } } { \mathrm { day } }$
D) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 1.2 \pi \frac { \mathrm { cm } } { \mathrm { day } }$
E) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 0.2 \pi \frac { \mathrm { cm } } { \mathrm { day } }$

Question

A mold culture in a dorm refrigerator is circular and growing. The radius is increasing at a rate of 0.1 cm/day. How fast is the area growing when the culture is 6 centimeters in radius  (The area of a disc of radius r is $A = \pi r ^ { 2 }$ .)  
A) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 18 \pi \frac { \mathrm { cm } } { \mathrm { day } }$
B)  $\frac { \mathrm { d } A } { \mathrm {~d} t } = 0.6 \pi \frac { \mathrm { cm } } { \mathrm { day } }$
C)   $\frac { \mathrm { d } A } { \mathrm {~d} t } = 2.4 \pi \frac { \mathrm { cm } } { \mathrm { day } }$
D) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 1.2 \pi \frac { \mathrm { cm } } { \mathrm { day } }$
E) $\frac { \mathrm { d } A } { \mathrm {~d} t } = 0.2 \pi \frac { \mathrm { cm } } { \mathrm { day } }$

Quizplus · Accepted Answer

The rate of change of area with respect to time is given by $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$. Substituting $r = 6$ cm and $\frac{dr}{dt} = 0.1$ cm/day, we get $\frac{dA}{dt} = 2\pi \times 6 \times 0.1 = 1.2\pi$ cm$^2$/day.

A Mold Culture in a Dorm Refrigerator Is Circular and Growing