the r.c. helliot advertising company finds that its profits on day t of an advertising campaign is given by…

the r.c. helliot advertising company finds that its profits on day t of an advertising campaign is given by $p(t)=-3t^{2}+60t + 40,000$ where $p(t)$ is profit in dollars for day t of the campaign. a) find a simplified expression for the marginal profit function. (be sure to use the proper variable in your answer.) $mp(t)=$ b) what is the exact rate of change of profits when 13 days have been passed in the campaign? the rate of change is dollars per day. (your answer may be negative.) (c) what are their total profits on day 13 of the campaign? total profits on day 13 are

the r.c. helliot advertising company finds that its profits on day t of an advertising campaign is given by $p(t)=-3t^{2}+60t + 40,000$ where $p(t)$ is profit in dollars for day t of the campaign. a) find a simplified expression for the marginal profit function. (be sure to use the proper variable in your answer.) $mp(t)=$ b) what is the exact rate of change of profits when 13 days have been passed in the campaign? the rate of change is dollars per day. (your answer may be negative.) (c) what are their total profits on day 13 of the campaign? total profits on day 13 are

Answer

Explanation:

Step1: Recall marginal profit formula

Marginal profit $MP(t)$ is the derivative of profit function $P(t)$. Given $P(t)=- 3t^{2}+60t + 40000$, use power - rule $\frac{d}{dt}(at^{n})=nat^{n - 1}$.

Step2: Differentiate each term

For the first term $-3t^{2}$, $\frac{d}{dt}(-3t^{2})=-3\times2t=-6t$. For the second term $60t$, $\frac{d}{dt}(60t)=60$. For the third term $40000$ (a constant), $\frac{d}{dt}(40000) = 0$. So, $MP(t)=-6t + 60$.

Step3: Find rate of change at $t = 13$

Substitute $t = 13$ into $MP(t)$. $MP(13)=-6\times13 + 60=-78 + 60=-18$.

Step4: Find total profit at $t = 13$

Substitute $t = 13$ into $P(t)$. $P(13)=-3\times(13)^{2}+60\times13 + 40000=-3\times169+780 + 40000=-507+780 + 40000=40273$.

Answer:

A. $MP(t)=-6t + 60$ B. $-18$ C. $40273$