Question

At a canning company the daily production cost, y, is given by the quadratic equation y = 650 – 15x + 0.45x2, where x is the number of canned items. What is the MINIMUM daily production cost?
A) $1,025.00 B) $1,186.25 C) $525.00 D) $536.25

Answer 1

The equation is a parabola so the minimum point is the vertex.

(-b/2a, y)

x = 15/0.9 = 16.67 canned itemz
y = 524.95 $

So it's C.

Answer 2

Answer:

Option C) is correct

Step-by-step explanation:

Given :  y is the daily production cost at a canning company such that $y=650-15x+0.45x^2$ wherex is the number of canned items .

To find : Minimum daily production cost

Solution :

$y=650-15x+0.45x^2$

On differentiating both sides with respect to x, we get

$\frac{\mathrm{d} y}{\mathrm{d} x}=-15+0.9x$

On putting $\frac{\mathrm{d} y}{\mathrm{d} x}=0$ , we get $-15+0.9x=0\Rightarrow 15=0.9x\Rightarrow x=\frac{15}{0.9}=\frac{150}{9}=\frac{50}{3}$

We get intervals as \left ( -\infty , \frac{50}{3}\right )\,,\,\left ( \frac{50}{3},\infty  \right )$\left ( -\infty , \frac{50}{3}\right )\,,\,\left ( \frac{50}{3},\infty  \right )$

For $x=0\epsilon \left ( -\infty ,\frac{50}{3} \right )$ , $f'(0)=-15< 0$

For $x=16\epsilon \left ( \frac{50}{3},\infty  \right )$ , $f'(18)=-15+0.9(18)=-15+16.2=1.2> 0$

Therefore, $y=\frac{50}{3}$ is a point of minima.

So, minimum cost is equal to $y\left ( \frac{50}{3} \right )$

$y\left ( \frac{50}{3} \right )=650-15\left ( \frac{50}{3} \right )+0.45\left ( \frac{50}{3} \right )^2=650-250+125=\ 525$

So, option C) is correct .