Question

HW 3 Linear Function Models: Problem 9

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(3 points)
A new highway has just opened and tolls will be collected from vehicles using the highway. The toll will be 119 cents for cars which travel 20 miles of the
highway. The toll will be 184 cents for cars which travel 33 miles of the highway. Let T(d) be a function which gives the total toll (in cents) which will be
charged to a car which travels d miles along the highway. Assume T(d) is a linear function. Find T'(d).
T(d) =
What will be the toll to travel 96 miles?
cents

Answer 1

• T(d) = 5d +19
• T(96) = 499 . . . cents

Explanation:

Given two points on the linear toll function T(d) are (20, 119) and (33, 184), you want the equation for the function T(d), and the value of T(96).

Slope

It is usually convenient to write the equation of a line in slope-intercept form when you want to find the value at a particular point. To use that form, we need to know the slope of the line. It is given by the formula ...

m = (y2 -y1)/(x2 -x1)

Using the given points, we have ...

m = (184 -119)/(33 -20) = 65/13 = 5

Point-slope equation

The point-slope form of the equation for a line is ...

y -k = m(x -h) . . . . . line with slope m through point (h, k)

Using the first point and the slope we found, the equation is ...

y -119 = 5(x -20)

Slope-intercept equation

Solving the above equation for y, we get ...

y = 5x -100 +119 . . . . . . . simplify, add 119

y = 5x +19 . . . . . . . . . . combine terms

Using variables d and T instead of x and y, the function is ...

T(d) = 5d +19

Toll for 96 miles

The value T(96) is ...

T(96) = 5(96) +19 = 480 +19 = 499

The toll for 96 miles is 499 cents.