Maximizing Volume and Revenue: Cubic Function Challenges

A box with a square base and height is to be constructed. If the volume of the box is given by the function V(x) = x^2(10 - x), where x is the length of the side of the base, what is the maximum volume of the box?

• 120.5

• 148.14814814814815

• 160.75

• 135.0

The revenue R from selling x units of a product is modeled by the function R(x) = -2x^3 + 30x^2 + 50. How many units should be sold to maximize revenue?

• 5.75

• 2.50

• 3.16

• 4.00

A water tank has a cubic shape, and its volume is given by V(s) = s^3, where s is the length of a side. If the tank is filled to a height of 5 meters, what is the volume of water in the tank?

• 100 cubic meters

• 75 cubic meters

• 150 cubic meters

• 125 cubic meters

The height of a projectile launched into the air is modeled by the function h(t) = -4.9t^3 + 20t^2 + 5, where t is time in seconds. What is the maximum height reached by the projectile?

• h(0) = 5

• h(2) = 25

• h(1.054) = 15.5

• h(1.5) = 18

A farmer wants to create a rectangular field with a cubic fence. The volume of the fence is given by V(x) = 3x^3 + 12x^2 + 9x. What is the critical point that maximizes the volume of the fence?

• 0

• 2

• -3

• -1.5

The profit P from producing x items is given by P(x) = -x^3 + 15x^2 - 24x. What is the maximum profit that can be achieved?

• 36.08

• 48.50

• 24.00

• 30.75

A car's speed is modeled by the function S(t) = 2t^3 - 15t^2 + 36t, where t is time in seconds. At what time does the car reach its maximum speed?

• 3 seconds

• 5 seconds

• 1 second

• 2 seconds

The volume of a balloon is modeled by the function V(r) = (4/3)πr^3, where r is the radius. If the radius increases from 2 cm to 5 cm, by how much does the volume increase?

• 100π cm³

• 200π cm³

• 50π cm³

• 156π cm³

A cylindrical water tank has a volume given by V(h) = πr^2h, where r is the radius and h is the height. If the radius is fixed at 3 meters, what is the height when the volume is 27π cubic meters?

• 4 meters

• 2 meters

• 5 meters

• 3 meters

The length of a side of a cube is increasing at a rate of 2 cm/s. If the side length is currently 4 cm, how fast is the volume of the cube increasing at that moment?

• 120 cm^3/s

• 72 cm^3/s

• 96 cm^3/s

• 48 cm^3/s