Chapter 8.3, Problem 20E

a.

To determine

To find:

The intercepts of theequation’s graph. Then graph the equation.

Answer to Problem 20E

x -intercept: $(9,0)$

y -intercept: $(0,18)$

Explanation of Solution

Given:

At the start of a trip, you fill up your car’s fuel tank with gas. After you drive x hours, the amount y (in gallons) of gas remaining is given by the equation $y=18-2x$ .

Calculation:

To find x -intercept, we will substitute $y=0$ in our given equation and solve for x as shown below:

  $\begin{array}{l}y=18-2x\left(\text{Given equation}\right)\\ 0=18-2x\left(\text{Substituting }y=0\right)\\ 0+2x=18-2x+2x\left(\text{Adding }2x\text{ on both sides}\right)\\ 2x=18\\ \frac{2x}{2}=\frac{18}{2}\left(\text{Dividing both sides by }2\right)\\ x=9\end{array}$

Therefore, the x -intercept of our given equation is $(9,0)$ .

To find y- intercept, we will substitute $x=0$ in our given equation and solve for y as shown below:

  $\begin{array}{l}y=18-2x\left(\text{Given equation}\right)\\ y=18-2(0)\left(\text{Substituting }x=0\right)\\ y=18-0\\ y=18\end{array}$

Therefore, the y -intercept of our given equation is $(0,18)$ .

Upon graphing our given equation, we will get our required graph as shown below:

  

b.

To determine

To interpret

What real-life quantitiesdo the x - and y -interceptsrepresent in this situation.

Answer to Problem 20E

The x -intercept of our given equation tell us that the tank of car will be empty after driving for 9 hours.

The y -intercept of our given equation tell us that at the start of the journey there were 18 gallons of fuel available in the tank.

Explanation of Solution

Given:

At the start of a trip, you fill up your car’s fuel tank with gas. After you drive x hours, the amount y (in gallons) of gas remaining is given by the equation $y=18-2x$ .

Calculation:

The x -intercept of our given equation tell us that the tank of car will be empty after driving for 9 hours. After driving for 9 hours the tank of the car will be empty.

The y -intercept of our given equation tell us that at the start of the journey there were 18 gallons of fuel available in the tank.

c.

To determine

After how many hours of driving do you have only $\frac{1}{4}$ tank of gas left.

Answer to Problem 20E

After 6.75 hours of driving wewill have only $\frac{1}{4}$ tank of gas left.

Explanation of Solution

Given:

At the start of a trip, you fill up your car’s fuel tank with gas. After you drive x hours, the amount y (in gallons) of gas remaining is given by the equation $y=18-2x$ .

Calculation:

To find the number of hours it will take the gas to be left $\frac{1}{4}$ of its original amount, we will substitute $y=\frac{18}{4}$ in our given equation and solve for x as:

  $\begin{array}{l}y=18-2x\left(\text{Given equation}\right)\\ \frac{18}{4}=18-2x\left(\text{Substituting }y=\frac{18}{4}\right)\\ \frac{18}{4}-18=18-18-2x\left(\text{Subtracting }18\text{from both sides}\right)\\ -13.5=-2x\\ -2x=-13.5\left(\text{Switching sides}\right)\\ \frac{-2x}{-2}=\frac{-13.5}{-2}\left(\text{Dividing both sides by }-2\right)\\ x=6.75\end{array}$

Therefore, after 6.75 hours of driving we will have only $\frac{1}{4}$ tank of gas left.