Let's go through the questions one by one, providing the answers and calculations where necessary.
1. Height of the Right Circular Cone
- Response: 15 cm
- Explanation: The height of the cone is given directly as 15 cm.
2. Surface Area of an Oblique Cylinder
The formula for surface area \(S\) of a cylinder is given by: \[ S = 2 \pi r^2 + 2 \pi r h \] Where:
- \( r = 5 \) cm (radius)
- \( h = 15 \) cm (height)
Computing: \[ S = 2 \times 3.14 \times (5^2) + 2 \times 3.14 \times 5 \times 15 \] \[ S = 2 \times 3.14 \times 25 + 2 \times 3.14 \times 5 \times 15 \] \[ S = 157 + 471 \] \[ S = 628 \text{ square centimeters} \]
- Response: 628 square centimeters
3. Surface Area of a Cylinder (Adrian's Coffee Table)
Using \( r = 4 \) feet and \( h = 3 \) feet: \[ S = 2 \pi r^2 + 2 \pi r h \] \[ S = 2 \times 3.14 \times (4^2) + 2 \times 3.14 \times 4 \times 3 \] \[ S = 2 \times 3.14 \times 16 + 2 \times 3.14 \times 12 \] \[ S = 100.48 + 75.36 \] \[ S = 175.84 \] Rounded to the nearest tenth: \[ S = 175.8 \text{ ft}^2 \]
4. Surface Area of a Cone (Castle Roof)
The formula for a cone's surface area is: \[ S = \pi r^2 + \pi r l \] Where:
- Diameter = 14 inches → \( r = 7 \) inches
- Slant height \( l = 20 \) inches
Computing: \[ S = \pi (7^2) + \pi (7)(20) \] \[ S = \pi (49) + \pi (140) \] \[ S = 189\pi \]
So surface area rounded to the nearest hundredth: \[ S \approx 593.86 \]
5. Volume of the Cylinder
Using \( r = 8 \) mm and \( h = 13 \) mm: \[ V = \pi r^2 h = \pi (8^2)(13) \] \[ V = \pi (64)(13) = 832\pi \] Approximation gives: \[ V \approx 2617.92 \text{ mm}^3 \]
6. Volume of the Water Bottle
Using \( r = 4 \) inches and \( h = 16 \) inches: \[ V = \pi r^2 h = \pi (4^2)(16) \] \[ V = \pi (16)(16) = 256\pi \] Approximately gives: \[ V \approx 804.25 \text{ in}^3 \]
7. Volume of Cone
Using \( r = 13 \) inches and \( h = 27 \) inches: \[ V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (13^2)(27) \] \[ V = \frac{1}{3}\pi (169)(27) = \frac{4563}{3}\pi = 1521\pi \] Approximated gives about \( 4777.89 \) cubic inches.
8. Volume of Cone-shaped Container
Using \( r = 4 \) feet, \( h = 12 \) feet:
- Formula: \[ V = \frac{1}{3}\pi r^2 h \] Calculating: \[ V = \frac{1}{3} \pi (4^2)(12) = \frac{1}{3} \pi (16)(12) = \frac{1}{3} \pi (192) = 64\pi \] Approximation \( V \approx 201.06 \) cubic feet.
9. Diameter of a Sphere
- Response: is the longest distance on a sphere.
10. Surface Area of a Hemisphere
Using \( r = 10 \) m: \[ S = 2\pi r^2 + \pi r^2 = 3\pi r^2 = 3\pi (10^2) = 300\pi \]
- Response: 300π m²
11. Surface Area of a Sphere
Given \( 7.84\pi = 4\pi r^2 \): \[ r^2 = \frac{7.84}{4} = 1.96 \rightarrow r = \sqrt{1.96} \approx 1.4 \text{ ft} \]
12. Volume of Ice Cream Scoop
Using \( r = 3 \) cm: \[ V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (3^3) = \frac{4}{3}\pi (27) = 36\pi \approx 113.04 \] Rounded to the nearest whole number: 113 cm³.
13. Relationship between Volumes
The correct relationship is:
- Response: 1 cylinder = 2 cones + 1 sphere.
14. Volume of Cylinder
Assuming the sphere has volume \( 48 , \text{m}^3 \) and diameter matches the cylinder's height:
- Use \( r \) from the sphere to find the cylinder \( V = \pi r^2 \cdot 2r = 2\pi r^3 \).
15. Total Volume in and on Top of Cone
If the ice cream cone's volume \( V = \frac{1}{3} \pi r^2 h \) and \( r = h \):
- Calculate the total volume by using previously derived relationships (consider cone and hemisphere).
This provides a comprehensive review of these geometry problems. Let me know if you would like detailed calculations for any specific item or further clarification!
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