Chapter 11.6, Problem 49STP

(a)

To determine

Write expressions for the time spent at that speed and for the time spent on the other highway.

Answer to Problem 49STP

  $\frac{150}{x};\frac{130}{x-10}$

Explanation of Solution

Given:

A trip between two towns takes 4 hours under ideal conditions. The first 150 miles of the trip is on an interstate, and the last 130 miles is on a highway with a speed limit that is 10 miles per hour less than on the interstate.

If a represents the speed limit on the interstate, write expressions for the time spent at that speed and for the time spent on the other highway.

Concept Used:

  $\text{Time}\text{taken}\text{=}\frac{\text{Distance}}{\text{Speed}}$

Distance traveled on the interstate is 150 miles.

Calculation:

  $\text{Time}\text{taken}\text{=}\frac{\text{Distance}}{\text{Speed}}$

Distance traveled on the interstate is 150 miles.

So, the time spent on the interstate is $\frac{150}{x}$

Distance traveled on highways is 130 miles. Speed on high way is $x-10$

So, the time spent on the highways is $\frac{130}{x-10}$

Thus, the time spent on the highways is $\frac{130}{x-10}$

(b)

To determine

Write and solve an equation to find the speed limits on the two highways.

Answer to Problem 49STP

  $\frac{150}{x}+\frac{130}{x-10}=4$; 75 mph; 65 mph

Explanation of Solution

Given:

A trip between two towns takes 4 hours under ideal conditions. The first 150 miles of the trip is on an interstate, and the last 130 miles is on a highway with a speed limit that is 10 miles per hour less than on the interstate.

Write and solve an equation to find the speed limits on the two highways.

Concept Used:

Total time taken for the trip is 4 hours.

The required equation: $\frac{150}{x}+\frac{130}{x-10}=4$

Calculation:

The required equation: $\frac{150}{x}+\frac{130}{x-10}=4$

Solve for x:

  $\begin{array}{l}150(x-10)+130x=4x(x-10)\\ 150x-1500+130x=4x^2-40x\\ 4x^2-40x-150x-130x+1500=0\\ 4x^2-320x+1500=0\\ x^2-80x+375=0\\ (x-5)(x-75)=0\\ x-5=0\Rightarrow x=5\\ x-75=0\Rightarrow x=75\end{array}$

So, the speed on the interstate is 75 mph.

Thus, the speed on the highway is 75 − 10 = 65 mph