Question

Type the correct answer in each box. (____)

A cylindrical hole is drilled into a piece of wood. The radius of the hole is 8 millimeters less than the height, x, of the hole.

If 1,024π cubic millimeters of material was removed to make the hole, the equation x^3 + ____ x^2 + ____ x − 1,024 = 0 can be used to find that the height of the hole is ____ millimeters.

______________________________

blanks:
1. ____ ?
2. ____ ?
3. ____ ?

Answer 1

1. -16; 2. +64; 3. 16

Explanation:

The formula for the volume of a cylinder is

V = πr²x

Data:

r = (x - 8) mm

V = 1024π mm³

Calculations:

1. Find the cubic equation

V = πr²h

1024π = π(x - 8)² × x

Divide each side by π

1024 = x(x - 8)²

1024 = x(x² - 16x + 64)

1024 = x³ - 16x² + 64x

x³ - 16x² + 64x - 1024 = 0

2. Solve the cubic equation

The general formula for a third-degree polynomial is

f(x) = ax³ + bx² + cx + d

Your polynomial is

f(x) = x³ - 16x² + 64x - 1024

a = 1; d = -1024

According to the rational roots theorem, the possible roots are

factors of d/factors of a

Factors of d = ±1, ±2, ±4, ±8, ±16, ± 32, ± 64, ±128, ±256, ±512, ±1024

Factors of a = ±1

Potential roots are x = ±1, ±2, ±4, ±8, ±16, ± 32, ± 64, ±128, ±256, ±512, ±1024

That's a lot of possibilities to check by trial and error. I will just use the one that works.

Try x = 16 by synthetic division.

16|1 -16 64 -1024

| 16 0 1024

1 0 64 0

So,

(x³ - 16x² + 64x - 1024)/(x – 16) = x² + 64

and

(x - 16)(x² + 64) = 0

x - 16 = 0 x² + 64 = 0

x = 16 x² = -64

x = ±8i

There is only one real root: x = 16 mm