Question

1. while playing his latest video game, mr. henderson notices that he is feeding the enemy team in such a way that the number of feeds can be calculated with the following formula: $f = \frac{\pi m^3}{m + 1}$, where m is the number of minions on the screen and f is the total number of feeds. mr. henderson knows that both the number of feeds and the number of minions are changing with time. furthermore, he observes that there are currently 9 minions on the screen and the number of minions is increasing by 2 minions per second. at this very moment in the game, what rate is mr. henderson feeding the enemy team?

Explanation:

Step1: Differentiate F with respect to t

Use quotient rule: $\frac{dF}{dt} = \pi \cdot \frac{(3m^2 \frac{dm}{dt})(m+1) - m^3 (\frac{dm}{dt})}{(m+1)^2}$
Simplify: $\frac{dF}{dt} = \pi \cdot \frac{m^2 \frac{dm}{dt}(3(m+1) - m)}{(m+1)^2} = \pi \cdot \frac{m^2 \frac{dm}{dt}(2m + 3)}{(m+1)^2}$

Step2: Substitute values

Given $m=9$, $\frac{dm}{dt}=2$:
$\frac{dF}{dt} = \pi \cdot \frac{9^2 \cdot 2 \cdot (2(9)+3)}{(9+1)^2} = \pi \cdot \frac{81 \cdot 2 \cdot 21}{100} = \pi \cdot \frac{3402}{100} = 34.02\pi$

Answer:

$34.02\pi$ feeds per second (or $\frac{1701}{50}\pi$)