BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 1.7, Problem 116E
To determine

The range of speed for which the vehicle’s mileage is 30mi/gal or more.

Expert Solution

Answer to Problem 116E

The vehicle’s mileage is 30mi/gal or more when the speed is in the interval [40,50]_.

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 g=10+0.9v0.01v2, where v is in between 10 mi/h and 75 mi/h.

Calculation:

To obtain the range of speed for which the vehicle’s mileage is 30mi/gal or more, set g30 and solve this inequality.

That is, solve the inequality 10+0.9v0.01v230.

Simplify the above inequality as follows.

10+0.9v0.01v2300[Subtract 30 from both sides]0.9v0.01v2200[Subtract 32 from both sides]v290v+20000[Divide by 0.01on both sides](v50)(v40)0[Factor the LHS]

Now, the factors of the left-hand side are v50 and v40.

These factors are zero when x is 50and 40, respectively.

These numbers divide the number line into three intervals: (,40),(40,50)and (50,).

Choose a test point from each of these intervals to determine the sign of the factors v50 and v40.

Consider the point 0 in the interval (,40). Then, v50 becomes 050=50 and has negative sign. And v40 becomes 040=40 and has negative sign. Then, the sign of the product (v50)(v40) will be positive.

Similarly, find the signs for the other two intervals and complete the table as follows.

Interval(,40)(40,50)(50,)
Sign of v50+
Sign of v40++
Sign of (v50)(v40)++

From the above table, observe that (v50)(v40)0 on the interval (40,50).

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.

(v50)(v40)?0(4050)(4040)?0(10)(0)?000[True]

Consider the end point 50.

(v50)(v40)?0(5050)(5040)?0(0)(10)?000[True]

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 [40,50]_.

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