Suppose you drive a car 392 mi on one tank of gas. The tank...

Suppose you drive a car 392 mi on one tank of gas. The tank holds 14 gallons. The number of miles traveled varies directly with the number of gallons of gas you use. a. Write an equation that relates miles traveled to gallons of gas used. b. You only have enough money to buy 3.7 gallons of gas. How far can you drive before refueling? c. Last year you drove $11,700$ mi. About how many gallons of gas did you use? d. Suppose the price of gas averaged $\$ 1.57$ per gallon last year. Find the cost per mile.

Suppose you drive a car 392 mi on one tank of gas. The tank holds 14 gallons. The number of miles traveled varies directly with the number of gallons of gas you use.
a. Write an equation that relates miles traveled to gallons of gas used.
b. You only have enough money to buy 3.7 gallons of gas. How far can you drive before refueling?
c. Last year you drove $11,700$ mi. About how many gallons of gas did you use?
d. Suppose the price of gas averaged $\$ 1.57$ per gallon last year. Find the cost per mile.

Key Concepts

Direct Variation

Direct variation describes a relationship where one quantity is a constant multiple of another. In this context, the number of miles traveled increases in direct proportion to the number of gallons of gas used, meaning the relationship can be expressed with an equation of the form miles = k × gallons.

Constant of Proportionality (Rate)

The constant of proportionality in a direct variation equation represents the rate at which one variable changes relative to another. Here, it is the car's fuel efficiency measured in miles per gallon (mpg), obtained by dividing the total miles traveled by the gallons of gas used.

Proportional Reasoning

Proportional reasoning involves setting up and solving ratios or proportions to find unknown values. This is used to compute distances achievable with a certain amount of gas or to determine the amount of gas used when given a total distance, relying on the constant rate of fuel efficiency.

Cost per Unit Analysis

Cost per unit analysis involves calculating the cost associated with a single unit of output, such as determining the cost per mile by dividing the total cost (derived from the price per gallon and total gallons used) by the total miles driven. This concept is helpful for budgeting and efficiency assessments.

Transcript

0:00 Problem.

00:01 We want to relate miles traveled to gallons of gas used.

00:06 And so we know that this is going to be a direct variation between miles and gallons.

00:13 And so essentially we can say that we're told that we can travel 392 miles on one tank of gas.

00:19 So that means that since one tank of gas holds 14 gallons, we can travel 392 miles with 14 gallons of gas.

00:29 And so to set up our direct relation, or to set up our equation, we can find our constant of variation by dividing our distance by our a gallon.

00:41 So i'll go ahead and write that as d over g.

00:43 So when we plug in 392 for our distance and 14 for our 14 gallons of gas, we're going to find that our constant of variation is equal to 28.

00:55 And so to set up our equation, we know that our equation is going to be our distance is equal to 28 times our gallons of gas.

01:05 And essentially all i did to get this was multiply both sides of this equation by g.

01:11 And so i got d is equal to k times g.

01:16 So i just had to substitute in my 28 for k.

01:22 So now that we have our expression here, our equation, we can answer part b.

01:27 So it says that we only have enough money to buy 3 .7 gallons of gas, and we want to know how far we can drive before refueling.

01:38 So essentially, we have 3 .7 gallons, and this is going to be equal to x amount of miles.

01:44 So we're going to be able to plug this into our equation that we just solved for.

01:50 So when we do that, we know that we're solving for d.

01:55 Which is our x miles, so i'll go ahead and actually just write that as x.

01:59 And this is equal to 28 times our gallons of gas, which we know is 3 .7.

02:05 And so when we multiply this out, we are going to find that we are able to drive 103 .6 miles before refueling for part c of our problem.

02:21 It tells us that we drove a distance of 11 ,700...

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