Question

Charles owns a watch repair shop. He has found that the cost of operating his shop is given by the quadratic function C(x) = 4x2 -368x + 53 where c is cost and x is the number of watches repaired. How many watches must he repair to have the lowest cost?

Answer 1

`\large \boxed{46 }`

Step-by-step explanation:

y = 4x² - 368x + 53

It would be easy to find the minimum using calculus, but we can also do it using algebra.

The standard form of the equation for a parabola is

y =ax² + bx + c

a = 4; b = -368; c = 53

The vertex is the point at which the parabola crosses its axis of symmetry. At that point,

`x = -(b)/(2a) = -(-368)/(2*4) = (368)/(8) = \mathbf{46}\\\\\text{Charles must repair at least $\large \boxed{\textbf{46 watches}}$ to have the lowest cost.}`

The graph below shows that Charles' cost is a minimum at 46 watches.

Answer 2

x = 46 watches

Step-by-step explanation:

If

C(x) = 4x² -368x + 53

then we apply

C(x)' = 0 ⇒ (4x² -368x + 53 )' = 8x - 368 = 0

⇒ x = 368 / 8 = 46 watches