How many watches must he repair to have the lowest cost?

Question

Bob owns a watch repair shop. He has found that the cost of operating his shop is given by c(x)=2x(squared)-112x+58, where c is the cost and x is the number of watches repaired.How many watches must he repair to have the lowest cost?

Answer 1

c(x) = 2x² - 112x + 58

 

When finding the lowest cost, we want to find the minimum of c(x). Since we have a positive parabola, the graph will open upward, indicating a minimum.

 

There are two ways to do this:

 

1) Take the derivative of c(x) and set it equal to zero.

2) Write c(x) in vertex form.

 

Since you are probably not up to derivatives yet, let's go to method 2.

 

The vertex form is

 

c(x) = a(x - h)² + k

 

where:

a = coefficient of x² term

 

and the vertex is in the form (h, k)

 

 

c(x) = 2(x² - 56x + 784) - 1510

c(x) = 2(x - 28)(x - 28) - 1510

c(x) = 2(x - 28)² - 1510

 

From this form,

 

h = 28

k = -1510

 

The vertex is  (28, -1510)

 

The coordinate is also the minimum.

 

Therefore, Bob must repair 28 watches to have the lowest cost.

Answer 2

C(x) = 2x² - 112x + 58

 

This is a quadratic equation whose graph is a parabola.  Because the coefficient of the x² term (2) is positive, we know that it open upwards with the vertex at the bottom.  The vertex is thus the lowest point of the graph, representing the lowest cost.

 

For a quadratic equation of the standard form y = ax² + bx + c, the vertex is always located at the point x = -b/2a.  For C(x), a = 2 and b = -112.  The vertex is thus located at:

 

x = -(-112)/4 = 28

 

Thus x = 28 watches will produce the minimum cost.