Chapter 8.1, Problem 56PPS

a.

To determine

Find the distance of ratio.

Answer to Problem 56PPS

The distance of ratio is $\frac{\left(500\right)\left(6+x\right)}{\left(540+90x\right)\left(6\right)}$ .

Explanation of Solution

Calculation:

Given formula:

  $\begin{array}{l}d=rt\\ =\left(500\right)\left(6+x\right)\text{ substitute the rate and time into the formula}\\ =\left(540+90x\right)\left(6\right)\\ =\frac{\left(500\right)\left(6+x\right)}{\left(540+90x\right)\left(6\right)}\text{ write as a ratio}\end{array}$

Thus, the distance of ratio is $\frac{\left(500\right)\left(6+x\right)}{\left(540+90x\right)\left(6\right)}$ .

b.

To determine

Simplify the rational expression.

Answer to Problem 56PPS

The rational expression is $25<27$

Explanation of Solution

Calculation:

From part (a)

  $\begin{array}{l}\frac{\left(500\right)\left(6+x\right)}{\left(540+90x\right)\left(6\right)}\\ =\frac{\left(500\right)\left(6+x\right)}{90\left(6+x\right)\cdot 6}\text{ like terms cancel out}\\ =\frac{500}{540}\text{ simplify}\\ =\frac{25}{27}\\ =25<27\text{ the second airplane travel a little faster than the first airplane}.\end{array}$

Thus, the rational expression is $25<27$ .

c.

To determine

Find the determine the volume of the region in terms of $x$ .

Answer to Problem 56PPS

The volume of the region in terms of $x=-6$

Explanation of Solution

Calculation:

Use the formula above to obtain:

  $\begin{array}{l}\frac{\left(500\right)\left(6+x\right)}{\left(540+90x\right)\left(6\right)}\\ \left(540+90x\right)\left(6\right)=0\\ 540+90x=0\text{ divided by }6\\ 90x=-540\text{ substract}540\\ x=-6\text{ divide }90\text{ when }x=-6\end{array}$

Thus, volume of the region in terms of $x=-6$