Question

The rate at which customers arrive at a counter to be served is modeled by the function $F$ defined by $F\left(t\right)=12+6\cos \left(\frac{t}{\pi }\right)$ for $0\le t\le 60$, where $F\left(t\right)$ is measured in customers per minute and $t$ is measured in minutes. To the nearest whole number, how many customers arrive at the counter over the $60-$minute period?

• A. $720$
• B. $725$
• C. $732$
• D. $744$
• E. $756$

Answer 1

The correct option is B $725$

Given : $F\left(t\right)=12+6\cos \left(\frac{t}{\pi }\right)$ Per minute

Number of customer

${\int }_0^{60}F\left(t\right)dt={\int }_0^{60}[12+6\cos \left(\frac{t}{\pi }\right)]dtN=12{\int }_0^{60}dt+6{\int }_0^{60}\cos \left(\frac{t}{\pi }\right)dt=12(60-0)+6\pi {\left[\sin \left(\frac{t}{\pi }\right)\right]}_0^{60}N=720+6\pi \left[\sin \left(\frac{60}{\pi }\right)\right]=720+5=725$

Hence the correct answer is $725$