Chapter 7.1, Problem 29E

To determine

To estimate: The total number of hot dogs sold during the first three hours using trapezoidal approximation.

Answer to Problem 29E

  $1800$

Explanation of Solution

Given information:

The table for rate of hot dogs sold for the first three hours.

Calculation:

Let $R\left(t\right)$ be the rate per minute at which the hot dogs were sold.

Then, the total number of hot dogs sold for the first three hours can be given as:

  $\text{Hot dogs sold}={\displaystyle \underset{0}{\overset{180}{\int }}R\left(t\right)dt}$ .

Approximate this integral using Trapezoidal rule for the given table using $n=10$ .

  $\begin{array}{c}{\displaystyle \underset{0}{\overset{180}{\int }}R\left(t\right)dt}=\left(\frac{3+6}{2}\cdot 10+\frac{6+18}{2}\cdot 20+\frac{18+16}{2}\cdot 15+\frac{16+12}{2}\cdot 15+\frac{12+8}{2}\cdot 30+\frac{8+10}{2}\cdot 30+\frac{10+6}{2}\cdot 60\right)\\ =45+240+255+210+300+270+480\\ =1800\end{array}$

Conclusion:

The total number of hot dogs sold over a period of 3-hours is $1800$ .