Application of a cubic. please help!

a. 28.

b 20.78.

Step-by-step explanation:

a. f(x) = 0.001(x -  10)(x + 20)(x - 40)

When x = 50

Height = f(40) = 0.001(50-10)(50+20)(50-40)

= 0.001(40*70*10)

= 0.001 * 28,000

= 28 units.

b.

Expanding the function we get

f(x) = 0.001(x  - 10)(x^2 - 20x - 800)

f(x) = 0.001(x^3 - 30x^2 - 600x + 8000)

f(x) = 0.001x^3 - 0.03x^2 - 0.6x + 8

Finding the derivative

f'(x) = 0.003x^2 - 0.06x - 0.6

This is zero for maxm/minm values:

0.003x^2 - 0.06x - 0.6 = 0

x = 27.32, - 7.32.

The values of the heights corresponding to these values of x are:

0.001(27.32-10)(27.32+20)(27.32-40)

= -10.39. and

0.001(-7.32-10)(-7.32+20)(-7.32-40)

= 10.39.

So the distance between A and B =  10.39 - (-10.39)

= 20.78.