Question

6x^3 + 110x^2 − 200x = 15000 write and solve
𝑦=____𝑥^3+____𝑥^2+___𝑥,
𝑦=___;𝑥= cm
What should the dimensions of the package be to have the maximum volume?

Answer 1

Fill in the blanks and solve:

y = 6x^3 + 110x^2 − 200x
y = 15000

Solve 6x^3 + 110x^2 − 200x = 15000:
Move 15000 to left: 6x^3 + 110x^2 − 200x − 15000 = 0.
Test x = 10: 6(10)^3 + 110(10)^2 − 200(10) − 15000 = 6000 + 11000 − 2000 − 15000 = 0, so (x − 10) is a factor.
Divide to get (x − 10)(6x^2 + 170x + 1500) = 0. The quadratic has negative discriminant, so the only real solution is
x = 10 cm.

To answer “What should the dimensions of the package be to have the maximum volume?” I need the expressions for the package dimensions in terms of x (length, width, height). Once you give those, just substitute x = 10 cm to get the numeric dimensions. If you post the three dimension formulas I’ll compute them for you.