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QuestionFind the cost equation c(d)c(d) for owning a car based on dd km/month. Predict cost for 10001000 km/month and analyze intercepts and slope.

Studdy Solution

STEP 1

Assumptions1. The cost of owning a car varies linearly with the distance driven per month. . The cost is 366permonthfordriving300kmpermonth.<br/>3.Thecostis366 per month for driving300 km per month.<br />3. The cost is 510 per month for driving1500 km per month.

STEP 2

We are given two points on the line that describes the cost as a function of distance (300,366) and (1500,510). We can use these points to find the slope of the line. The slope (m) is given by the formulam=y2−y1x2−x1 m = \frac{y2 - y1}{x2 - x1}

STEP 3

Substitute the given points into the formula for the slope.
m=510−3661500−300 m = \frac{510 -366}{1500 -300}

STEP 4

Calculate the slope.
m=1441200=0.12 m = \frac{144}{1200} =0.12

STEP 5

Now that we have the slope, we can use the point-slope form of a line to write the equation for the cost as a function of distance. The point-slope form is given byy−y1=m(x−x1) y - y1 = m(x - x1)

STEP 6

Substitute the slope and one of the given points (300,366) into the point-slope form.
c−366=0.12(d−300) c -366 =0.12(d -300)

STEP 7

implify the equation to get it into slope-intercept form (y = mx + b).
c=0.12d+b c =0.12d + b

STEP 8

To find the y-intercept (b), substitute one of the given points into the equation and solve for b.
366=0.12⋅300+b366 =0.12 \cdot300 + b

STEP 9

Calculate the y-intercept.
b=366−.12⋅300=300 b =366 -.12 \cdot300 =300

STEP 10

Now we have the equation for the cost as a function of distance.
c=0.12d+300 c =0.12d +300

STEP 11

To predict the monthly cost for driving1000 km/month, substitute1000 for d in the equation.
c=0.⋅1000+300 c =0. \cdot1000 +300

STEP 12

Calculate the monthly cost.
c=0.12⋅1000+300=$420 c =0.12 \cdot1000 +300 = \$420

STEP 13

The cost-rate intercept is the y-intercept of the equation, which is 300.Thismeansthatthebasecostofowningthecar,withoutdrivingitatall,is300. This means that the base cost of owning the car, without driving it at all, is 300 per month.

STEP 14

To check the prediction, substitute the predicted cost into the equation and solve for d.
420=0.12d+300420 =0.12d +300

STEP 15

olve for d.
d=420−3000.12=1000 km/month d = \frac{420 -300}{0.12} =1000 \, km/month

STEP 16

The distance intercept is the x-intercept of the equation, which is found by setting c equal to0 and solving for d.
0=0.12d+3000 =0.12d +300

STEP 17

olve for d.
d=−3000.12=−2500 km/month d = \frac{-300}{0.12} = -2500 \, km/month

STEP 18

The slope of the equation is0.12. This means that for each additional kilometer driven per month, the cost of owning the car increases by $0.12.

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