SOLUTION: The volume of gas in your car’s gas tank depends on the number of miles you have driven. Suppose you leave home with a full tank and begin driving at a constant speed. After driv

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Question 1104227: The volume of gas in your car’s gas tank depends on the number of miles you have driven.
Suppose you leave home with a full tank and begin driving at a constant speed. After driving 120 miles you have 12 gallons in your tank.
After driving 220 miles you have 8.66 gallons in your tank.
Assume the relationship is linear.

a. Find a function that models this situation. (Input miles driven and output gallons of gas in your tank.)

b. How many gallons does your gas tank hold when full?

c. How many miles can you drive before you will run out of gas?

d. What is your car's approximate fuel efficiency in miles per gallon?

Found 4 solutions by josmiceli, ikleyn, josgarithmetic, Theo:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +y+ = gallons of gas in tank
Let +x+ = miles driven
---------------------------------
The more miles that are driven, the
less gallons are in the tank
The general form for a linear equation is:
+y+=+m%2Ax+%2B+b+
--------------------------------------
(1) +12+=+m%2A120+%2B+b+
(2) +8.66+=+m%2A220+%2B+b+
--------------------------------
Subtract (2) from (1)
(1) +12+=+m%2A120+%2B+b+
(2) +-8.66+=+-m%2A220+-+b+
-------------------------------
+3.34+=+-100m+
+m+=+-.0334+
and
(1) +12+=+%28+-.0334+%29%2A120+%2B+b+
(1) +12+=+-4.008++%2B+b+
(1) +b+=+16.008+
---------------------------------
(a) +y+=+-.0334x+%2B+16.008+
---------------------------------
check:
+y+=+-.0334%2A120+%2B+16.008+
+y+=+-4.008+%2B+16.008+
+y+=+12+
and
+y+=+-.0334%2A220+%2B+16.008+
+y+=+-7.348+%2B+16.008+
+y+=+8.66+
OK
---------------------------------
(b)
When +x+=+0+ miles driven, the tank is full
+y+=+-.0334%2A0+%2B+16.008+
There are 16.008 gallons in tank when full
-------------------------------------------
(c)
When does +y+=+0+ ?
+0+=+-.0334x+%2B+16.008+
+.0334x+=+16.008+
+x+=+479.281+
I can drive 479.281 mi before running out of gas
-----------------------------------------------
(d)
[ gallons in tank ] = [ -gallons/mi ] + [ gallons w/full tank ]
+y+=+m%2Ax+%2B+b+
+m+=+-.0334+
[ miles / gallon ] = +1%2F.0334+
+1%2F.0334+=+29.94+
The fuel efficiency is 29.94 mi/gal

Answer by ikleyn(52770) About Me  (Show Source):
You can put this solution on YOUR website!
The volume of gas in your car’s gas tank depends on the number of miles you have driven. Suppose you leave home with a full tank
and begin driving at a constant speed. After driving 120 miles you have 12 gallons in your tank. After driving 220 miles
you have 8.66 gallons in your tank. Assume the relationship is linear. a. Find a function that models this situation.
(Input miles driven and output gallons of gas in your tank.) 


According to the condition, the car spends 12-8.66 = 3.33 gallons per 220-120 = 100 miles,  or 3.33%2F100 = 0.0333 gallons per mile, 
or,  which is the same, 1%2F0.0333 = 30 miles per gallon.
 
 
b. How many gallons does your gas tank hold when full?    It holds 12 + 0.0333*120 = 15.996  = about  16  gallons when full.
 
 
 
c. How many miles can you drive before you will run out of gas?  16%2F0.0333 = 480 miles (approximately).
 


d. What is your car’s approximate fuel efficiency in miles per gallon?   30 miles per gallon, as I calculated it above.


Answer by josgarithmetic(39615) About Me  (Show Source):
You can put this solution on YOUR website!
Application for a linear equation.
Your data points for ordered pair are x for miles driven, and y for volume of fuel in the tank.

The two points are (120, 12) and (220, 8.66).
Find the equation of the line with these two points.

%2812-8.66%29%2F%28120-220%29=3.34%2F%28-100%29=-0.0334, slope

y-12=-0.0334%28x-120%29

y=-0.0334%28x-120%29%2B12

y=-0.0334x%2B12%2B4.008
highlight%28y=-0.0334x%2B16.008%29

Full tank holds 16 gallons of fuel. (If you want to three significant figures).

Fuel Efficiency was 100%2F3.34=30, THIRTY miles per gallon, (if to three significant figures, then 30.0 mpg).

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
let x equal your miles.
let y equal the number of gallons of gas.

it's a linear relationship.
the slope intercept form of a linear equation is y = mx + b
m is the slope
b is the y-intercept.

you have 2 coordinates you can work with.
they are (120,12) and (220,8.66)

let the first one be (x1,y1).
let the second one be (x2,y2).

m = slope = (y2-y1) / (x2-x1) = (8.66-12)) / (220-120) = -.0334
equation of y = mx + b becomes y = -.0334 * x + b

use either one of the points to find the y-intercept.
i'll use (120,12).
replace y with 12 and x with 120 to get 12 = -.0334 * 120 + b
solve for b to get b = 16.008

the equation becomes y = -.0334 * x + 16.008

when x = 0, you have a full tank.
when x = 120, you have -.0334 * 120 + 16.008 = 12
when x = 220, you have -.0334 * 220 + 16.008 = 8.66

to find out when the tank will be empty, set y = 0 to get 0 = -.0334 * x + 16.008.
subtract 16.008 from both sides of this equation to get -16.008 = -.0334 * x.
divide both sides of this equation by .0334 to get -16.008 / .0334 = x.
solve for x to get x = -16.008 / .0334 = 479.2814371.

the tank will be empty after the car has driven 479.2814371 miles.

the equation can be graphed.
the graph looks like this:

$$$

since you tank holds 16.008 gallons of gas and you can travel 499.2814371 miles before you run out of gas, then your gas mileage is equal to 479.2814371 / 16.008 = 29.94011976 miles per gallon.

that's approximately equal to 30 miles per gallon based on 16 gallon capacity and 480 miles range.