Concept explainers
Gas Mileage The gas mileage g (measured in mi/gal) for a particular vehicle, driven at v mi/h. is given by the formula g = 10 + 0.9v − 0.01v2, as long as v is between 10 mi/h and 75 mi/h. For what range of speeds is the vehicle’s mileage 30 mi/gal or better?
The range of speed for which the vehicle’s mileage is 30mi/gal or more.
Answer to Problem 116E
The vehicle’s mileage is 30mi/gal or more when the speed is in the interval
Explanation of Solution
Given:
The gas mileage g (measured in mi/gal) for a particular vehicle, driven at v mi/h, is given by the formula
Calculation:
To obtain the range of speed for which the vehicle’s mileage is 30mi/gal or more, set
That is, solve the inequality
Simplify the above inequality as follows.
Now, the factors of the left-hand side are
These factors are zero when
These numbers divide the number line into three intervals:
Choose a test point from each of these intervals to determine the sign of the factors
Consider the point
Similarly, find the signs for the other two intervals and complete the table as follows.
Interval | |||
Sign of | |||
Sign of | |||
Sign of |
From the above table, observe that
That is, the inequality is satisfied by the points from 40 to 50.
Now, check whether the endpoints satisfy the inequality.
Consider the end point 40.
Consider the end point 50.
Thus, the endpoints are included in the solution.
Then, the solution will be a closed interval.
Hence, the vehicle’s mileage is 30mi/gal or more when the speed is in the interval
Chapter 1 Solutions
Precalculus: Mathematics for Calculus - 6th Edition