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. -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 rational roots theorem, the possible roots arefactors 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