Answer:
1. -16; 2. +64; 3. 16
Step-by-step 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 <em>rational roots theorem</em>, 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
<u>| 16 0 1024
</u>
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
and 
. Get everything on one side of the equals sign, set it equal to 0 and factor. When you do this you get (x-3)(x+27). The Zero Product Property rule tells us that either x-3 = 0 or x+27 = 0 and that x = 3 and -27. The only thing in math that will NEVER be negative besides time is distance/length, therefore, x cannot be 27 and has to be 3.