Question

1 a car has a 16 - gallon fuel tank. when driven on a highway, it has a gas mileage of 30 miles per gallon. the gas mileage (also called “fuel efficiency”) tells us the number of miles the car can travel for a particular amount of fuel (one gallon of gasoline, in this case). after filling the gas tank, the driver got on a highway and drove for a while. a. how many miles has the car traveled if it has the following amounts of gas left in the tank? i. 15 gallons ii. 10 gallons iii. 2.5 gallons b. write an equation that represents the relationship between the distance the car has traveled in miles, d, and the amount of gas left in the tank in gallons, x. c. how many gallons are left in the tank when the car has traveled the following distances on the highway? i. 90 miles ii. 246 miles d. write an equation that makes it easier to find the amount of gas left in the tank, x, if we know the car has travelled d miles. 2 the area a of a rectangle is represented by the formula a = lw where l is the length and w is the width. the length of the rectangle is 5. write an equation that makes it easy to find the width of the rectangle if we know the area and the length.

Explanation:

Step1: Calculate fuel - used

For part a, first find the amount of fuel used by subtracting the remaining fuel from the full - tank capacity (16 gallons). Then multiply the fuel used by the gas mileage (30 miles per gallon) to get the distance traveled.
For part b, let the full - tank capacity be 16 gallons, the amount of gas left be $x$, so the fuel used is $16 - x$. The distance $d$ is the product of the fuel used and the gas mileage.
For part c, first find the fuel used by dividing the distance traveled by the gas mileage, then subtract the fuel used from the full - tank capacity to get the remaining fuel.
For part d, solve the equation from part b for $x$.

Part a

i.

Step1: Calculate fuel used

The full - tank capacity is 16 gallons and the remaining fuel is 15 gallons. So the fuel used is $16−15 = 1$ gallon.
$16−15=1$

Step2: Calculate distance traveled

Since the gas mileage is 30 miles per gallon, the distance traveled $d=30\times1 = 30$ miles.
$d = 30\times1=30$

ii.

Step1: Calculate fuel used

The remaining fuel is 10 gallons. So the fuel used is $16 - 10=6$ gallons.
$16−10 = 6$

Step2: Calculate distance traveled

The distance traveled $d = 30\times6=180$ miles.
$d=30\times6 = 180$

iii.

Step1: Calculate fuel used

The remaining fuel is 2.5 gallons. So the fuel used is $16 - 2.5 = 13.5$ gallons.
$16−2.5=13.5$

Step2: Calculate distance traveled

The distance traveled $d=30\times13.5 = 405$ miles.
$d=30\times13.5=405$

Part b

The full - tank capacity is 16 gallons. If the amount of gas left in the tank is $x$ gallons, the amount of gas used is $(16 - x)$ gallons. Since the gas mileage is 30 miles per gallon, the distance $d$ traveled is given by the equation $d = 30(16 - x)$.

Part c

i.

Step1: Calculate fuel used

The gas mileage is 30 miles per gallon and the distance traveled is 90 miles. The fuel used is $\frac{90}{30}=3$ gallons.
$\text{Fuel used}=\frac{90}{30}=3$

Step2: Calculate remaining fuel

The full - tank capacity is 16 gallons. So the remaining fuel $x=16 - 3 = 13$ gallons.
$x=16−3=13$

ii.

Step1: Calculate fuel used

The gas mileage is 30 miles per gallon and the distance traveled is 246 miles. The fuel used is $\frac{246}{30}=8.2$ gallons.
$\text{Fuel used}=\frac{246}{30}=8.2$

Step2: Calculate remaining fuel

The full - tank capacity is 16 gallons. So the remaining fuel $x=16 - 8.2 = 7.8$ gallons.
$x=16−8.2=7.8$

Part d

Starting with the equation $d = 30(16 - x)$ from part b, we first expand it: $d=480-30x$. Then we solve for $x$:

Step1: Rearrange the equation

$30x=480 - d$

Step2: Solve for $x$

$x=\frac{480 - d}{30}$

Part 2

Given the area formula of a rectangle $A=lw$ and $l = 5$. To find the width $w$ when we know the area $A$ and the length $l$, we solve the equation $A=lw$ for $w$.

Step1: Isolate $w$

Divide both sides of the equation $A = 5w$ by 5.
$w=\frac{A}{5}$

Answer:

a.
i. 30 miles
ii. 180 miles
iii. 405 miles
b. $d = 30(16 - x)$
c.
i. 13 gallons
ii. 7.8 gallons
d. $x=\frac{480 - d}{30}$

1. $w=\frac{A}{5}$