Cross Sections, Solids of Rotation and Geometric Modeling: Geometry Math Test – PDF Download

Question 1: Name the cross sections (3 points)

Name the shape of each cross section.

1. (1 point) The cut through the cylinder.
2. (1 point) The cut through the cone.
3. (1 point) The cut through the sphere.

Question 2: Cut through a cylinder (2 points)

A cylinder has a diameter of \(12\text{ cm}\) and a height of \(15\text{ cm}\). A plane passes through its axis perpendicular to the bases.

1. (1 point) Name the shape of the cross section and give its dimensions.
2. (1 point) Find the area of the cross section.

Question 3: Slicing a sphere (2 points)

A sphere has a radius of \(10\text{ cm}\). A plane cuts it \(6\text{ cm}\) from the center.

1. (1 point) Find the radius of the circular cross section.
2. (1 point) Find the area of the cross section in terms of \(\pi\).

Question 4: Rotating a triangle (3 points)

A right triangle has legs of \(5\text{ cm}\) and \(12\text{ cm}\). It is rotated about the \(12\text{ cm}\) leg.

1. (1 point) Name the solid.
2. (1 point) Give its radius and height.
3. (1 point) Find its volume in terms of \(\pi\).

Question 5: Comparing counties (2 points)

County A has 31,200 residents on \(40\text{ mi}^{2}\). County B has 24,640 residents on \(28\text{ mi}^{2}\).

1. (1 point) Find the population density of each county.
2. (1 point) Which county is more densely populated, and by how many people per square mile?

Question 6: Heating a gym (2 points)

A gym is \(50\text{ ft}\) long, \(30\text{ ft}\) wide and \(12\text{ ft}\) tall. It needs 4 BTUs per cubic foot to be heated.

1. (1 point) How many BTUs are needed?
2. (1 point) A heater supplies \(18{,}000\) BTUs per hour. How many hours does it take?

Question 7: Cheaper box (3 points)

Noah needs a box that holds \(360\text{ in}^{3}\). Box 1 is \(10\text{ in}\) by \(6\text{ in}\) by \(6\text{ in}\). Box 2 is \(9\text{ in}\) by \(8\text{ in}\) by \(5\text{ in}\). Cardboard costs \(\$0.03\) per square inch.

1. (1 point) Show that both boxes hold \(360\text{ in}^{3}\).
2. (1 point) Find the surface area of each box.
3. (1 point) Find which box costs less to make and its cost.

Question 8: Grain silo (3 points)

A silo is a cylinder with a radius of \(5\text{ m}\) and a height of \(12\text{ m}\), topped by a hemisphere with a radius of \(5\text{ m}\).

1. (1 point) Find the volume of the cylinder in terms of \(\pi\).
2. (1 point) Find the volume of the hemisphere in terms of \(\pi\).
3. (1 point) Find the total volume to the nearest tenth.

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