Examkin

Question

Difficulty: MediumVolume and Surface Area of Solids

A container in the shape of a right circular cylinder with a base radius of 44 inches and a height of 99 inches is completely filled with water. All of the water from this cylinder is poured into an empty container in the shape of a rectangular prism. The rectangular prism container has a square base of side length 66 inches and a height of 1515 inches. If the height of the water in the rectangular prism container is kπk\pi inches, what is the value of kk?

Answer: 4

Answer

The value of kk is 44.
To find the height of the water in the rectangular prism, we first calculate the volume of the water using the cylinder's volume formula, V=πr2hV = \pi r^2 h. With a radius of 44 inches and a height of 99 inches, the volume is π(4)2(9)=144π\pi (4)^2 (9) = 144\pi cubic inches. When this water is poured into the rectangular prism, the volume of the water can also be represented as the area of the base times the height of the water: V=Base Area×hw=62×hw=36hwV = \text{Base Area} \times h_w = 6^2 \times h_w = 36 h_w. Equating the two volumes gives 36hw=144π36 h_w = 144\pi, which simplifies to hw=4πh_w = 4\pi inches. Since the height of the water in the prism is kπk\pi inches, the value of kk is 44.

Step-by-Step Solution

1
Calculate the volume of the water in the filled cylinder using the formula V=πr2hV = \pi r^2 h.
The volume of the water is 144π144\pi cubic inches.
Since the cylinder is completely filled, the volume of the water is equal to the volume of the cylinder with base radius 44 inches and height 99 inches.
2
Calculate the area of the square base of the rectangular prism container.
The base area is 3636 square inches.
The base of the prism is a square with side length 66 inches, so its area is 6×6=366 \times 6 = 36 square inches.
3
Set the volume of the water in the rectangular prism equal to the volume of the water from the cylinder, and solve for the water height hwh_w.
36×hw=144π  ⟹  hw=4π36 \times h_w = 144\pi \implies h_w = 4\pi inches.
Pouring the water into the prism container does not change its volume, which remains 144π144\pi cubic inches. The volume of a prism is the base area times its height.
4
Compare the height of the water 4π4\pi to the expression kπk\pi to find kk.
k=4k = 4.
Since the height of the water is represented as kπk\pi inches and we calculated it to be 4π4\pi inches, kk must equal 44.

Key Concept

Equating the volumes of a right circular cylinder and a rectangular prism to solve for an unknown dimension.
Rate this question

Topics

Volume and Surface Area of Solids

More questions from this topic