could you please help me answer this

Question

it costs $ 90 to rent a car driven 100 miles and $140 for one driven 200 miles . if x is the number of miles driven and y the total cost of the rental write the cost function.

Answer 1

Hi Matt,

 

I'll be happy to help you understand this problem.

 

The cost function will be a linear function represented by the equation of a line.  Since two points determine a line, we can write the equation of the line given our two points.

 

The equation of a line in slope-intercept form is:

    y = mx + b (m is slope and b is y-intercept)

 

We are told that x is the number of miles and y is the total rental cost.  Therefore, our two given points (x₁,y₁) and (x₂,y₂) are (100,90) and (200,140), respectively.

 

The slope (m) is given by the formula:

    m = (y₂ - y₁) / (x₂ - x₁)

 

Plugging in our values for (x₁,y₁₎ and (x₂,y₂), we get:

   m = (140 - 90) / (200 - 100)

   m = 50/100

   m = 1/2

 

To determine the y-intercept (b) which is where the line crosses the y-axis, we can plug in our slope and one of our given points into the slope-intercept form of the equation for a line and solve it for b:

    y = mx + b (linear equation:  slope-intercept form)

    m = 1/2 (our slope, determined above)

    x = 100 (x₁)

    y = 90 (y₁)

    90 = (1/2)(100) + b

    b = 90 - (1/2)(100)

    b = 90 - 50

    b = 40

 

Now we have everything we need to write our cost function, y (output cost), in terms of x (input miles):

    y = mx + b (linear equation:  slope-intercept form)

    slope (m) = 1/2

    y-intercept (b) = 40

    y = (1/2)x + 40

    y = x/2 + 40 (cost function)

 

The key to our understanding of this problem lies in the fact that if we are given two points on a line then we can determine the slope and y-intercept of the line and write it's equation in slope-intercept form.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.