Question

1. find the volume of these solids, rounding your answers to three decimal places where necessary.

1. find the missing lengths, correct to one decimal place.

Explanation:

Here are the step-by-step solutions for all the problems in the image.

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🆕 New Concept Discovered: Volume of 3D Solids
Calculating space inside pyramids, spheres, cones, and prisms.

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Part 7: Find the volume of these solids

Round answers to three decimal places where necessary.

Step 1: Top-Left Solid (Pyramid with given base area)

The formula for the volume of a pyramid is:

$$ V = \frac{1}{3} \times \text{Base Area} \times \text{height} $$

Given:

• \( \text{Base Area } (A) = 50\text{ cm}^2 \)
• \( \text{height } (h) = 30\text{ cm} \)

Calculation:

$$ V = \frac{1}{3} \times 50 \times 30 = 500\text{ cm}^3 $$

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Step 2: Top-Middle Solid (Sphere)

The formula for the volume of a sphere is:

$$ V = \frac{4}{3}\pi r^3 $$

Given:

• \( \text{diameter} = 4.8\text{ m} \implies \text{radius } (r) = 2.4\text{ m} \)

Calculation:

$$ V = \frac{4}{3} \times \pi \times (2.4)^3 $$

$$ V = \frac{4}{3} \times \pi \times 13.824 \approx 57.906\text{ m}^3 $$

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Step 3: Top-Right Solid (Cone)

The formula for the volume of a cone is:

$$ V = \frac{1}{3}\pi r^2 h $$

Given:

• \( \text{diameter} = 1\text{ m} \implies \text{radius } (r) = 0.5\text{ m} \)
• \( \text{height } (h) = 1.2\text{ m} \)
• (Note: The slant height of \(1.3\text{ m}\) is extra information, but we can verify it using Pythagoras: \(\sqrt{0.5^2 + 1.2^2} = 1.3\))

Calculation:

$$ V = \frac{1}{3} \times \pi \times (0.5)^2 \times 1.2 $$

$$ V = \frac{1}{3} \times \pi \times 0.25 \times 1.2 = 0.1\pi \approx 0.314\text{ m}^3 $$

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Step 4: Bottom-Left Solid (Rectangular Pyramid)

The formula for the volume of a pyramid with a rectangular base is:

$$ V = \frac{1}{3} \times (\text{length} \times \text{width}) \times \text{height} $$

Given:

• \( \text{length} = 8\text{ cm} \)
• \( \text{width} = 5\text{ cm} \)
• \( \text{height } (h) = 27\text{ cm} \)

Calculation:

$$ V = \frac{1}{3} \times (8 \times 5) \times 27 $$

$$ V = \frac{1}{3} \times 40 \times 27 = 360\text{ cm}^3 $$

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Step 5: Bottom-Middle Solid (Hemisphere)

The formula for the volume of a hemisphere is half the volume of a sphere:

$$ V = \frac{2}{3}\pi r^3 $$

Given:

• \( \text{radius } (r) = 7\text{ cm} \)

Calculation:

$$ V = \frac{2}{3} \times \pi \times 7^3 $$

$$ V = \frac{2}{3} \times \pi \times 343 = \frac{686}{3}\pi \approx 718.378\text{ cm}^3 $$

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Part 8: Find the missing lengths

Round answers to one decimal place.

Step 6: Top-Left (Rectangular Prism)

The formula for the volume of a rectangular prism is:

$$ V = \text{length} \times \text{width} \times \text{height} $$

Given:

• \( V = 1134\text{ cm}^3 \)
• \( \text{length} = 14\text{ cm} \)
• \( \text{width} = x\text{ cm} \)
• \( \text{height} = x\text{ cm} \)

Set up the equation:

$$ 1134 = 14 \times x \times x $$

$$ 1134 = 14x^2 $$

$$ x^2 = \frac{1134}{14} = 81 $$

$$ x = \sqrt{81} = 9.0\text{ cm} $$

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Step 7: Top-Middle (Cylinder)

The formula for the volume of a cylinder is:

$$ V = \pi r^2 h $$

Given:

• \( V = 703\text{ cm}^3 \)
• \( \text{height } (h) = 12\text{ cm} \)

Set up the equation:

$$ 703 = \pi \times r^2 \times 12 $$

$$ r^2 = \frac{703}{12\pi} $$

$$ r^2 \approx \frac{703}{37.6991} \approx 18.6476 $$

$$ r = \sqrt{18.6476} \approx 4.3\text{ cm} $$

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Step 8: Top-Right (Sphere)

The formula for the volume of a sphere is:

$$ V = \frac{4}{3}\pi r^3 $$

Given:

• \( V = 972\pi\text{ cm}^3 \)

Set up the equation:

$$ 972\pi = \frac{4}{3}\pi r^3 $$

Divide both sides by \(\pi\):

$$ 972 = \frac{4}{3}r^3 $$

$$ r^3 = 972 \times \frac{3}{4} $$

$$ r^3 = 729 $$

$$ r = \sqrt[3]{729} = 9.0\text{ cm} $$

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Step 9: Bottom-Left (Right Triangular Prism)

T…

Answer:

Part 7:

1. Top-Left Pyramid: \(500\text{ cm}^3\)
2. Top-Middle Sphere: \(57.906\text{ m}^3\)
3. Top-Right Cone: \(0.314\text{ m}^3\)
4. Bottom-Left Pyramid: \(360\text{ cm}^3\)
5. Bottom-Middle Hemisphere: \(718.378\text{ cm}^3\)

Part 8:

1. Top-Left Prism: \(x = 9.0\text{ cm}\)
2. Top-Middle Cylinder: \(r = 4.3\text{ cm}\)
3. Top-Right Sphere: \(r = 9.0\text{ cm}\)
4. Bottom-Left Prism: \(x = 11.0\text{ cm}\)
5. Bottom-Middle Pyramid: \(x = 0.3\text{ m}\)
6. Bottom-Right Cone: \(h = 13.0\text{ cm}\)