Chapter 5.2, Problem 89AYU

a.

To determine

To find: Vehicle Stopping Distance: Taking into account reaction time, the distance d (in feet) that a car requires to come to a complete stop while traveling r miles per hour is given by the function.

  $d\left(r\right)=6.97r-90.39$

a. Express the speed r at which the car is traveling as a function of the distance d required to come to a complete stop.

Answer to Problem 89AYU

Solution:

a.The speed r at which the car is travelling as a function of the distance d.

Required to come to a complete stop is $r=r\left(d\right)=\frac{d+90.39}{6.97}$ .

Explanation of Solution

Given:

The distance function $d\left(r\right)=6.97r-90.39$ .

Calculation:

a.Given the distance function $d\left(r\right)=6.97r-90.39$ .

We can find the speed r in terms of d as

  $d=6.97r-90.39$

  $d+90.39=6.97 r$

  $6.97r=d+90.39$

  $r=r\left(d\right)=\frac{d+90.39}{6.97}$ .

b.

To determine

To find: Vehicle Stopping Distance: Taking into account reaction time, the distance d (in feet) that a car requires to come to a complete stop while traveling r miles per hour is given by the function.

  $d\left(r\right)=6.97r-90.39$

b. Verify that r = r(d) is the inverse of d = d(r) by showing that r(d(r)) = r and d(r(d)) = d.

Answer to Problem 89AYU

Solution:

b. Verified that $r=r\left(d\left(r\right)\right)$ and $\left(r\left(d\right)\right)=d$ .

Explanation of Solution

Given:

The distance function $d\left(r\right)=6.97r-90.39$ .

Calculation:

b. Verify $r=r\left(d\right)$ is the inverse $d=d\left(r\right)$ by showing below.

  $r\left(d\right)=r\left(d\left(r\right)\right)=r\left(6.97r-90.39\right)$

  $r\left(d\right)=\frac{6.97r-90.39+90.39}{6.97}$

  $r\left(d\right)=\frac{6.97r}{6.97}$

  $r\left(d\right)=r$

Similarly, we can prove

  $d=d\left(r\right)=d\left(r\left(d\right)\right)$

  $d\left(r\right)=d\left(\frac{d+90.39}{6.97}\right)$

  $d\left(r\right)=6.97\left(\frac{d+90.39}{6.97}\right)-90.39$

  $d\left(r\right)=d+90.39-90.39$

  $d\left(r\right)=d$

c.

To determine

To find: Vehicle Stopping Distance: Taking into account reaction time, the distance d (in feet) that a car requires to come to a complete stop while traveling r miles per hour is given by the function.

  $d\left(r\right)=6.97r-90.39$

c. Predict the speed that a car was traveling if the distance required to stop was 300 feet.

Answer to Problem 89AYU

Solution:

c. $r=56 \text{miles per hour}$ .

Explanation of Solution

Given:

The distance function $d\left(r\right)=6.97r-90.39$ .

Calculation:

c. The speed that a car was traveling if the distance required to stop was 300 feet.

We know that $r=r\left(d\right)=\frac{d+90.39}{6.97}$ .

  $d=300\text{ feet}$ .

Therefore, $r=\frac{300+90.39}{6.97}=\frac{390.39}{6.97}$ .

  $r=56.05308$ .

  $r=56 \text{miles per hour}$ .