# Estimating a Hemisphere

Warmup

Activity #1

Investigate the Relationship Between Volume of Sphere and Volume of a Hemisphere. • Interact with the applet below and answer the question that follows.

How does the volume of a sphere compare with the volume of a hemisphere with the same length radius?

Activity #2

Exercises on Solving a Hemisphere. Mai has a dome paperweight that she can use as a magnifier. The paperweight is shaped like a hemisphere made of solid glass, so she wants to design a box to keep it in so it won’t get broken. Her paperweight has a radius of 3 cm. • Study the diagram and answer the questions that follow. (1.) What should the dimensions of the inside of box be so the box is as small as possible?

(2.) What is the volume of the box?

(3.) What is a reasonable estimate for the volume of the paperweight?

Tyler has a different box with side lengths that are twice as long as the sides of Mai’s box. Tyler’s box is just large enough to hold a different glass paperweight.

(1.) What is the volume of the new box?

(2.) What is a reasonable estimate for the volume of this glass paperweight?

(3.) How many times bigger do you think the volume of the paperweight in this box is than the volume of Mai’s paperweight? Explain your thinking.

A hemisphere with radius 5 units fits snugly into a cylinder of the same radius and height. (1.) Calculate the volume of the cylinder.

(2.) Estimate the volume of the hemisphere. Explain your reasoning.

A cone fits snugly inside a hemisphere, and they share a radius of 5. (1.) What is the volume of the cone?

(2.) Estimate the volume of the hemisphere. Explain your reasoning.

(3.) Compare your estimate for the hemisphere with the cone inside to your estimate of the hemisphere inside the cylinder.

(4.) How do they compare to the volumes of the cylinder and the cone?

Activity #3

Exercise Questions on Solving a Hemisphere.  A hemisphere fits snugly inside a cylinder with a radius of 6 cm. A cone fits snugly inside the same hemisphere.

(1.) What is the volume of the cylinder?

(2.) What is the volume of the cone?

(3.) Estimate the volume of the hemisphere by calculating the average of the volumes of the cylinder and cone.

(4.) Find the hemisphere’s diameter if its radius is 6 cm.

(5.) Find the hemisphere’s diameter it its radius is 100/3 meters.

(6.) Find the hemisphere’s diameter if its radius is 9.008 ft.

(7.) Find the hemisphere’s radius if its diameter is 6 cm.

(8.) Find the hemisphere’s radius if its diameter is 100/3 meters.

(9.) Find the hemisphere’s radius if its diameter is 9.008 ft.

Challenge #1

A baseball fits snugly inside a transparent display cube. The length of an edge of the cube is 2.9 inches.

Is the baseball’s volume greater than, less than, or equal to 2.9³ cubic inches? Explain how you know.

Challenge #2

There are many possible cones with a height of 18 meters. Let r represent the radius in meters and V represent the volume in cubic meters.

(1.) Write an equation that represents the volume V as a function of the radius r.

(2.) Complete this table for the function, giving three possible examples.

Challenge #3

(1.) If you double the radius of a cone, does the volume double?

(a.) Explain how you know.

(b.) Is the graph of this function a line? Explain how you know.

Quiz Time