💡 7th Grade Math (Pre-Algebra): Cylinders Practice Questions

Here's how to find the volume:

• Identify the given values: Radius (\(r\)) = 5 cm, Height (\(h\)) = 10 cm.
• Use the volume formula: \( V = \pi r^2 h \)
• Substitute the values: \( V = \pi (5 \text{ cm})^2 (10 \text{ cm}) \)
• Calculate: \( V = \pi (25 \text{ cm}^2) (10 \text{ cm}) \)
• Simplify: \( V = 250\pi \text{ cm}^3 \)

The volume of the cylinder is \( 250\pi \) cubic centimeters. You can also approximate this by using \( \pi \approx 3.14 \), which gives \( V \approx 785 \) cm³.

Let's calculate the surface area step-by-step:

• Given values: Radius (\(r\)) = 3 inches, Height (\(h\)) = 7 inches.
• Apply the surface area formula: \( SA = 2\pi r^2 + 2\pi rh \)
• Substitute the values: \( SA = 2\pi (3 \text{ in})^2 + 2\pi (3 \text{ in})(7 \text{ in}) \)
• Calculate the area of the bases: \( 2\pi (9 \text{ in}^2) = 18\pi \text{ in}^2 \)
• Calculate the lateral surface area: \( 2\pi (21 \text{ in}^2) = 42\pi \text{ in}^2 \)
• Add them together: \( SA = 18\pi \text{ in}^2 + 42\pi \text{ in}^2 \)
• Simplify: \( SA = 60\pi \text{ in}^2 \)

The total surface area of the cylinder is \( 60\pi \) square inches. Approximately, this is \( 60 \times 3.14 \approx 188.4 \) in².

Here's how to solve this:

• Find the radius: Diameter = 6 cm, so Radius (\(r\)) = 6 cm / 2 = 3 cm.
• Given height: Height (\(h\)) = 10 cm.
• Use the volume formula: \( V = \pi r^2 h \)
• Substitute the values: \( V = \pi (3 \text{ cm})^2 (10 \text{ cm}) \)
• Calculate: \( V = \pi (9 \text{ cm}^2) (10 \text{ cm}) \)
• Simplify: \( V = 90\pi \text{ cm}^3 \)

The volume of the soup can is \( 90\pi \) cubic centimeters. This is approximately \( 90 \times 3.14 \approx 282.6 \) cm³.

Let's work backward to find the radius:

• Given values: Lateral Surface Area (\(LSA\)) = \( 100\pi \) m², Height (\(h\)) = 5 m.
• Use the LSA formula: \( LSA = 2\pi rh \)
• Substitute known values: \( 100\pi \text{ m}^2 = 2\pi r (5 \text{ m}) \)
• Simplify the right side: \( 100\pi \text{ m}^2 = 10\pi r \text{ m} \)
• Isolate \(r\): Divide both sides by \( 10\pi \text{ m} \).
• Calculate: \( r = \frac{100\pi \text{ m}^2}{10\pi \text{ m}} = 10 \text{ m} \)

The radius of the cylinder is 10 meters. ✅

Let's compare the volumes of both containers:

• Container A:
• Radius (\(r_A\)) = 4 cm
• Height (\(h_A\)) = 8 cm
• Volume (\(V_A\)) = \( \pi r_A^2 h_A = \pi (4 \text{ cm})^2 (8 \text{ cm}) = \pi (16 \text{ cm}^2)(8 \text{ cm}) = 128\pi \text{ cm}^3 \)
• Container B:
• Radius (\(r_B\)) = 8 cm
• Height (\(h_B\)) = 4 cm
• Volume (\(V_B\)) = \( \pi r_B^2 h_B = \pi (8 \text{ cm})^2 (4 \text{ cm}) = \pi (64 \text{ cm}^2)(4 \text{ cm}) = 256\pi \text{ cm}^3 \)

Comparing the volumes: \( V_B = 256\pi \text{ cm}^3 \) and \( V_A = 128\pi \text{ cm}^3 \).

Conclusion: Container B holds more volume because \( 256\pi > 128\pi \). Even though Container A is taller, the larger radius of Container B has a greater impact on its volume due to the \(r^2\) term in the formula.

Let's find the total area to be painted:

• Given values: Radius (\(r\)) = 10 ft, Height (\(h\)) = 20 ft.
• Use the total surface area formula: \( SA = 2\pi r^2 + 2\pi rh \)
• Substitute the values: \( SA = 2\pi (10 \text{ ft})^2 + 2\pi (10 \text{ ft})(20 \text{ ft}) \)
• Calculate the area of the bases: \( 2\pi (100 \text{ ft}^2) = 200\pi \text{ ft}^2 \)
• Calculate the lateral surface area: \( 2\pi (200 \text{ ft}^2) = 400\pi \text{ ft}^2 \)
• Add them together: \( SA = 200\pi \text{ ft}^2 + 400\pi \text{ ft}^2 \)
• Simplify: \( SA = 600\pi \text{ ft}^2 \)

The total surface area to be painted is \( 600\pi \) square feet. This is approximately \( 600 \times 3.14 \approx 1884 \) ft².

Let's calculate the capacity of the soup can:

• Given values: Radius (\(r\)) = 4 cm, Height (\(h\)) = 12 cm.
• Use the volume formula: \( V = \pi r^2 h \)
• Substitute the values: \( V = \pi (4 \text{ cm})^2 (12 \text{ cm}) \)
• Calculate: \( V = \pi (16 \text{ cm}^2) (12 \text{ cm}) \)
• Simplify: \( V = 192\pi \text{ cm}^3 \)

The soup can hold \( 192\pi \) cubic centimeters of soup. This is approximately \( 192 \times 3.14 \approx 602.88 \) cm³.

Let's calculate the frosting area:

• Given values: Radius (\(r\)) = 7 inches, Height (\(h\)) = 5 inches.
• Use the total surface area formula: \( SA = 2\pi r^2 + 2\pi rh \)
• Substitute the values: \( SA = 2\pi (7 \text{ in})^2 + 2\pi (7 \text{ in})(5 \text{ in}) \)
• Calculate the area of the bases: \( 2\pi (49 \text{ in}^2) = 98\pi \text{ in}^2 \)
• Calculate the lateral surface area: \( 2\pi (35 \text{ in}^2) = 70\pi \text{ in}^2 \)
• Add them together: \( SA = 98\pi \text{ in}^2 + 70\pi \text{ in}^2 \)
• Simplify: \( SA = 168\pi \text{ in}^2 \)

You need to frost a total surface area of \( 168\pi \) square inches. This is approximately \( 168 \times 3.14 \approx 527.52 \) in².