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

Answer:

x = 46 watches

Explanation:

If

C(x) = 4x² -368x + 53

then we apply

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

⇒   x = 368 / 8 = 46 watches

Answer 2

Answer:

$\large \boxed{46 }$

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,

$x = -\dfrac{b}{2a} = -\dfrac{-368}{2\times4} = \dfrac{368}{8} = \mathbf{46}\\\\\text{Charles must repair at least \large \boxed{\textbf{46 watches}} to have the lowest cost.}$

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