Question

practice problems
10 problems
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 initial fuel volume

The car has a 16 - gallon fuel tank.

Step2: Find fuel used

For part a, find the fuel used by subtracting the remaining fuel from the initial 16 gallons. Then multiply the fuel used by the mileage (30 miles per gallon) to get the distance traveled.

i.

Initial fuel = 16 gallons, remaining fuel = 15 gallons. Fuel used = 16 - 15=1 gallon. Distance traveled $d = 30\times(16 - 15)=30$ miles.

ii.

Initial fuel = 16 gallons, remaining fuel = 10 gallons. Fuel used = 16 - 10 = 6 gallons. Distance traveled $d=30\times(16 - 10)=180$ miles.

iii.

Initial fuel = 16 gallons, remaining fuel = 2.5 gallons. Fuel used = 16 - 2.5 = 13.5 gallons. Distance traveled $d = 30\times(16 - 2.5)=405$ miles.

Step3: Derive equation for part b

Let the distance traveled be $d$ and the amount of gas left in the tank be $x$. The fuel used is $16 - x$. Since the mileage is 30 miles per gallon, $d=30\times(16 - x)$.

Step4: Solve for remaining fuel in part c

For part c, first find the fuel used by dividing the distance traveled by the mileage (30 miles per gallon), then subtract the fuel used from the initial 16 gallons to get the remaining fuel.

i.

Distance traveled $d = 90$ miles. Fuel used $=\frac{90}{30}=3$ gallons. Remaining fuel $x=16 - 3 = 13$ gallons.

ii.

Distance traveled $d = 246$ miles. Fuel used $=\frac{246}{30}=8.2$ gallons. Remaining fuel $x=16 - 8.2 = 7.8$ gallons.

Step5: Rearrange equation for part d

Starting from $d = 30\times(16 - x)$, we can solve for $x$. First, divide both sides by 30: $\frac{d}{30}=16 - x$. Then, $x = 16-\frac{d}{30}$.
For the rectangle problem:
Given $A=lw$ and $l = 5$, to find $w$, we can rewrite the formula as $w=\frac{A}{5}$.

Answer:

a.
i. 30 miles
ii. 180 miles
iii. 405 miles
b. $d = 30\times(16 - x)$
c.
i. 13 gallons
ii. 7.8 gallons
d. $x = 16-\frac{d}{30}$
For rectangle: $w=\frac{A}{5}$