Chapter 11.3, Problem 53PPE

a.

To determine

To solve the formula $d=rt$ for t .

Answer to Problem 53PPE

  $t=\frac{d}{r}$

Explanation of Solution

Given information:

The formula for the distance traveled is $d=rt$

Concept used:

Exponent properties: If a , b are real numbers and m , n are whole numbers, then

Product Property: $a^m\cdot a^n=a^{m+n}$

Quotient property: $\frac{a^m}{a^n}=a^{m-n}$

Power Property: ${\left(a^m\right)}^n=a^{m\cdot n}$

Product to a Power Property: ${\left(ab\right)}^m=a^m{b}^m$

Calculation:

In order to solve the given formula for the variable t , divide both sides by r as shown below

  $\begin{array}{l}d=rt\\ \frac{d}{r}=\frac{rt}{r}\\ \frac{d}{r}=t\end{array}$

Thus, the formula for t is $t=\frac{d}{r}$ .

b.

To determine

To expression for the time it takes to travel a distance of $t^3-6t^2+5t+12$ miles at a rate of $t+1$ miles per hour.

Answer to Problem 53PPE

  $t^2-7t-2+\frac{14}{t+1}$

Explanation of Solution

Given information:

The formula for the distance traveled is $d=rt$

Concept used:

Exponent properties: If a , b are real numbers and m , n are whole numbers, then

Product Property: $a^m\cdot a^n=a^{m+n}$

Quotient property: $\frac{a^m}{a^n}=a^{m-n}$

Power Property: ${\left(a^m\right)}^n=a^{m\cdot n}$

Product to a Power Property: ${\left(ab\right)}^m=a^m{b}^m$

Calculation:

In order to find the expression for the time it take to travel the distance, use the long division method and divide the total distance by the speed expression as shown below,

  $t+1\begin{array}{c}\hfill t^2-7t-2\\ \hfill \overline{)\begin{array}{l}{t}^3-6t^2+5t+12\\ \underset{\_}{-\left(t^3+t^2\right)}\\ -7t^2+5t\\ \underset{\_}{-\left(-7t^2+7t\right)}\\ -2t+12\\ \underset{\_}{-\left(-2t-2\right)}\\ 14\end{array}}\end{array}$

Thus, the expression for the time taken is $t^2-7t-2+\frac{14}{t+1}$ .