PLEASE HELP ME!!!!!!!!!!

Question

An aircraft factory manufactures airplane engines. The unit cost C

 (the cost in dollars to make each airplane engine) depends on the number of engines made. If x

 engines are made, then the unit cost is given by the function C(x)=0.5x² - 280x +48,305

. What is the minimum unit cost?

Do not round your answer.

Answer 1

We do not have to use the quadratic formula to solve the problem

A. With knowledge of derivatives you are asking for the value of x that minimizes C(x).

C'(x) = [ .5x² -280 x + 48305 ]'

C'(x) = x -280

Set C'(x) = 0 to find the critical points

That is x = 280

But C'' (x) = 1

Since C'' (280) >0 then using the second derivative test the function C(x) has a minimum at x = 280

which is C(280) = 9105.

B. Without again using the quadratic formula since the cost function is in the form of a parabola

y = ax² +bx +c

with a = 0.5 >0 then the function has a minimum at x = -b/ 2a = 280

Answer 2

To solve for 'x", you would have to use the quadratic formula:

x = (-b + (b² - 4*a*c)^1/2) / 2*a

where a = 0.5, b = -280, and c = 48,305

x = (-(-280)+((-280)²-4*0.5*48,305)^1/2) / 2*0.5

Since you cannot have a negative number under the square root, that part of the equation is 0.

x = (-(-280)) / 1 = 280

If you plug 280 for x into C(280), then you have:

C(280) = 0.5 * (280)² - 280*280 + 48305 = $9,105