Larsen’s Sharpening Shop, specializes in sharpening saws. Terry Larsen has determined that the cost of operating the shop is given by the function:
C(x) = x^2 – 40x + 430, where x is the number of saws sharpened daily.
a. Write C(x) = x^2 - 40x + 430 = a(x-h)^2 + k, identify a, h, and k.
b. What is the number of saws Terry must sharpen each day in order to minimize the operational cost?
c. What is the cost under these ideal condition.
a. To rewrite the given function in the form C(x) = a(x-h)^2 + k, we need to complete the square.
C(x) = x^2 - 40x + 430
To complete the square, we need to find the values of a, h, and k.
First, let's focus on the quadratic term, x^2. We can rewrite it as (x - 20)^2 by adding and subtracting (40/2)^2 inside the parentheses:
C(x) = (x^2 - 40x + 400) + 30x + 430 - 400
= (x - 20)^2 + 30x + 30
So, a = 1, h = 20, and k = 30.
b. In order to minimize the operational cost, we need to find the vertex of the parabola, which corresponds to the lowest point on the graph.
The vertex of a parabola in the form y = a(x-h)^2 + k is given by the coordinates (h, k).
In this case, h = 20 and k = 30.
Therefore, Terry needs to sharpen 20 saws each day in order to minimize the operational cost.
c. To determine the cost under these ideal conditions, we substitute the value of h into the original function:
C(h) = h^2 - 40h + 430
= 20^2 - 40(20) + 430
= 400 - 800 + 430
= 30
So, the cost under these ideal conditions will be 30.