Find the number of cars that should use the car wash and the cost per hour for this number.

Question

The cost per hour of running a car wash is C(x)=0.04x² - 0.74x +5.75, where C(x) is the cost per hour in dollars and x is the number of cars passing through the car wash per hour. Determine the most economical number of cars that should use the car wash and the cost per hour for this number.

Answer 1

Chels,

 

The derivative of a function of one variable is its changing tangent or slope at different values of the independent variable x.  If you set that derivative to zero, you can find the turning point (or the actual value of x) for that function where it is either at its maximum or its minimum.  Thus a function that is rising as x increases reaches a point beyond which it starts to fall, and conversely, function that is falling as x increases reaches a point beyond which it starts to rise.  These unique points are the turning points for the function.  In general, linear functions have no turning points since they are either constant lines or ever increasing and decreasing by their slopes.  Hence their derivative is just the slope.  So the function must have powers of x higher than one to have bending or curvatures to look for turning points.

 

In your case,  C(x) = 0.04x² - 0.74x + 5.75  so it bends

Its derivative, d(C(x))/dx = 0.08x - 0.74 = 0  solves for x = 9.25  per hour

 

Now let's evaluate C(x) for several values of x around that solution:

C(5)      = 0.04(25) - 0.74(5) + 5.75 = $ 3.050/hour

C(9)      = 0.04(81) - 0.74(9) + 5.75 = $ 2.330/hour

C(9.25) = 0.04(85.5625) - 0.74(9.25) + 5.75 = $ 2.328/hour

C(10)    = 0.04(100) - 0.74(10) + 5.75 = $ 2.350/hour

C(15) = 0.04(225) - 0.74(15) + 5.75 = $ 3.650/hour

 

This verifies that this cost function has a minimum at x= 9.25 cars per hour.  You can either go with that as an average of cars per hour across a whole day, a week, month, etc, as businesses do in real life.  Or you can say x = 9 cars per hour, each hour, if you don't want fractional cars.  Rounded to the penny, the costs per hour are identical at $ 2.33