Let's tackle each question step-by-step:
1. **Height of the right circular cone**:
- The height of the right circular cone is 15 cm.
**Answer: 15 cm**
2. **Surface area of the oblique cylinder**:
- The surface area of a cylinder is found using the formula \(SA = 2\pi r(h + r)\).
- Here, \(r = 5 \text{ cm}\) and \(h = 15 \text{ cm}\).
\[
SA = 2 \cdot 3.14 \cdot 5 \cdot (15 + 5) = 2 \cdot 3.14 \cdot 5 \cdot 20 = 628 \text{ square centimeters}
\]
**Answer: 628 square centimeters**
3. **Surface area of the painted cylinder**:
- The surface area of the cylinder, including the top and bottom, is given by:
\[
SA = 2\pi r(h + r)
\]
- Here, \(r = 4 \text{ feet}\) and \(h = 3 \text{ feet}\).
\[
SA = 2 \cdot 3.14 \cdot 4 \cdot (3 + 4) = 2 \cdot 3.14 \cdot 4 \cdot 7 = 175.84 \text{ square feet}
\]
**Answer: 175.8 square feet**
4. **Surface area of the cone peak**:
- Surface area of the cone includes the base and the side (lateral surface):
\[
A = \pi rL + \pi r^2
\]
- Here, \(r = \frac{14}{2} = 7 \text{ inches}\) and \(L = 20 \text{ inches}\).
\[
A = 3.14 \cdot 7 \cdot 20 + 3.14 \cdot 7^2 = 439.6 + 153.86 = 593.46 \text{ square inches}
\]
**Answer: 593.46 square inches**
5. **Volume of the cylinder**:
- The volume of a cylinder is \(V = \pi r^2 h\).
- Here, \(r = 8 \text{ mm} \) and \(h = 13 \text{ mm} \).
\[
V = 3.14 \cdot 8^2 \cdot 13 = 3.14 \cdot 64 \cdot 13 = 2610.56 \text{ cubic millimeters}
\]
**Answer: 2610.56 cubic millimeters**
6. **Volume of the water bottle**:
\[
V = \pi r^2 h = 3.14 \cdot 4^2 \cdot 16 = 3.14 \cdot 16 \cdot 16 = 803.84 \text{ cubic inches}
\]
**Answer: 803.84 cubic inches**
7. **Volume of a cone**:
- The volume of a cone is \(V = \frac{1}{3} \pi r^2 h\).
- Here, \(r = 13 \text{ inches}\) and \(h = 27 \text{ inches}\).
\[
V = \frac{1}{3} \cdot 3.14 \cdot 13^2 \cdot 27 = \frac{1}{3} \cdot 3.14 \cdot 169 \cdot 27 = 4788.78 \text{ cubic inches}
\]
**Answer: 4788.78 cubic inches**
8. **Volume of the feed container**:
- The volume of the feed container is \(V = \frac{1}{3} \pi r^2 h\).
- Here, \(r = 4 \text{ feet}\) and \(h = 12 \text{ feet}\).
\[
V = \frac{1}{3} \cdot 3.14 \cdot 4^2 \cdot 12 = \frac{1}{3} \cdot 3.14 \cdot 16 \cdot 12 = 200.96 \text{ cubic feet}
\]
**Answer: 200.96 cubic feet**
9. **Definition of the diameter of a sphere**:
- The correct definition is that the diameter of a sphere is the longest distance on a sphere.
**Answer: is the longest distance on a sphere**
10. **Surface area of a hemisphere**:
- Surface area of a hemisphere is \(2\pi r^2 + \pi r^2 = 3\pi r^2\).
- Here, \(r = 10\).
\[
SA = 3 \cdot \pi \cdot 10^2 = 300\pi \text{ square meters}
\]
**Answer: 300π m²**
11. **Radius from surface area of the sphere**:
- The surface area of a sphere is \(4\pi r^2\).
\[
4\pi r^2 = 7.84\pi \quad \Rightarrow \quad r^2 = \frac{7.84}{4} = 1.96 \quad \Rightarrow \quad r = \sqrt{1.96} = 1.4 \text{ feet}
\]
**Answer: 1.4 feet**
12. **Volume of the spherical scoop of ice cream**:
- The volume of a sphere is \( \frac{4}{3}\pi r^3 \).
- Here, \(r = 3 \text{ cm}\).
\[
V = \frac{4}{3} \cdot 3.14 \cdot 3^3 = \frac{4}{3} \cdot 3.14 \cdot 27 = 113.04 \approx 113 \text{ cubic centimeters}
\]
**Answer: 113 cm³**
13. **Volume relationship between the sphere, cylinder, and cone**:
- Given the radius \(r\), height \(h = 2r\):
- Volume of cylinder: \( \pi r^2 h \)
- Volume of cone: \( \frac{1}{3} \pi r^2 h \)
- Volume of sphere: \( \frac{4}{3}\pi r^3 \)
\[
V_{\text{cylinder}} = 2 \cdot \text{Volume of cone}
\]
So indeed, \(3 \text{ cones} = 1 \text{ cylinder}\).
**Answer: 3 cones = 1 cylinder**
14. **Volume of the cylinder**:
- Sphere volume \( = \frac{4}{3}\pi r^3 = 48 \) m³
- Cylinder with same radius and height = sphere's diameter (\(2r\)):
\[
r^3 = \frac{36}{4} = 12 \quad \Rightarrow \quad r = \sqrt[3]{12}
\]
The volume of the cylinder:
\[
V_{\text{cylinder}} = \pi r^2 \cdot 2r = 2\pi r^3 = 2 \cdot 48 = 96 \text{ m³}
\]
**Answer: 96 m³**
15. **Total volume for the ice cream in the cone and hemisphere**:
- Volume of a hemisphere:
\[
V_{\text{hemisphere}} = \frac{4}{3} \pi \left(\frac{d}{2}\right)^3 = 4 \text{ in}^3
\]
This implies
\[
4 = \frac{2}{3}\pi r^3 \Rightarrow r = 1.588 \text{ cubic inches}
\]
Therefore,
\[
V_{\text{hemisphere}} = 2\pi r^2 =2 \cdot 4 = 8 \text{ in}^3
\]
Answer:
\[ = 4 +8 = 12 in^3
\]
Note down: pi of ice cream are typically assumed the canonical | analytical value would be typically take assumptions of general eqn to best fit.
**Answer: 16 in³**
Cones, Cylinders, & Spheres Unit Test
1. Use the image to answer the question.
A cone is marked with a height of 15 centimeters and a diameter of 16 centimeters, both shown as perpendicular dotted lines. The slant height is 17 centimeters.
What is the height of the right circular cone?
(1 point)
Responses
17 cm
17 cm
16 cm
16 cm
8 cm
8 cm
15 cm
2. Use the image to answer the question.
An oblique cylinder is labeled with a height of 15 centimeters and radius 5 centimeters.
Find the surface area of the cylinder below. Use the approximation 3.14 for pi.
(1 point)
Responses
533.8 square centimeters
533.8 square centimeters
628 centimeters
628 centimeters
1,099 square centimeters
1,099 square centimeters
628 square centimeters
628 square centimeters
3. Adrian is painting the outside of a cylinder that he plans to use as a coffee table. The cylinder has a radius of 4 feet and a height of 3 feet. Adrian wants to paint all around the outside of the cylinder, including the top and bottom faces. In order to understand how much paint is needed, he wants to know the surface are of the outside of the cylinder. What is the surface area of the cylinder, measured in square feet? Use 3.14 for pi and round your answer to the nearest tenth.(1 point)
ft2
4. Eli is making a model castle out of clay. One of the roof peaks is in the shape of a cone with a diameter of 14 inches and a slant height of 20 inches. What is the surface area of the cone peak? Round your answer to the nearest hundredth. Use 3.14 for pi.(1 point)
5. Use the image to answer the question.
A 3 D cylinder shows a base radius of 8 millimeters and perpendicular height of 13 millimeters. A right angle is formed at the center of the base.
Find the volume of the cylinder, in cubic millimeters. Round your answer to the nearest hundredth.
(1 point)
6. A water bottle has a height of 16 inches and a radius of 4 inches. What is the volume, in cubic inches, of the water bottle? Use 3.14 for pi. (1 point)
7. Find the volume, in cubic inches, of a cone with a radius of 13 inches and a height of 27 inches. Round your answer to the nearest hundredth. Use 3.14 for pi.(1 point)
8. A cone-shaped container on a farm is used for storing animal feed. The container has a radius of 4 feet and a height of 12 feet. What volume of feed, in cubic feet, can the container store? Round your answer to the nearest hundredth. Use 3.14 for π
.(1 point)
9. The diameter of a sphere(1 point)
Responses
is the longest distance on a sphere.
is the longest distance on a sphere.
is the distance from the center of the sphere to any point on the surface of the sphere.
is the distance from the center of the sphere to any point on the surface of the sphere.
does not pass through the center of the sphere.
does not pass through the center of the sphere.
is half the length of the radius.
10. Find the surface area of a hemisphere with a radius 10m. Leave your answer in terms of pi.(1 point)
Responses
500π m2
500 pi m squared
300π m2
300 pi m squared
400π m2
400 pi m squared
200π m2
11. The surface area of a sphere is 7.84π ft.2 What is the radius of this sphere?(1 point)
12. How much ice cream is in a spherical scoop of ice cream with a radius of 3 cm? Use 3.14 as an approximation for pi. Round your answer to the nearest whole number.(1 point)
13. Suppose a sphere, cylinder, and cone all share the same radius and the cylinder and cone also share the same height, which is twice the radius. Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone?(1 point)
Responses
1 cylinder = 2 cones + 1 sphere
1 cylinder = 2 cones + 1 sphere
3 cones = 1 cylinder
3 cones = 1 cylinder
sphere = cylinder + cone
sphere = cylinder + cone
sphere - cone = cylinder
14. The volume of a sphere is 48 m3 . What is the volume of a cylinder if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?(1 point)
15. Use the image to answer the question.
A hemisphere is placed on top of an inverted cone. The back of the circumference, the radius, and the perpendicular height of the cone are marked with dashed lines. The radius and the height make a right angle and are perpendicular.
Aimee packs ice cream into an ice cream cone. She then puts a perfect hemisphere of ice cream on top of the cone that has a volume of 4 in.3 The diameter of the ice cream cone is equal to its height. What is the total volume of ice cream in and on top of the cone? Use the relationship between the formulas for the volumes of cones and spheres to help solve this problem. Show your work and explain your reasoning.
3 answers
u sure?
Let's review the answers to ensure their accuracy:
1. **Height of the right circular cone**:
- As mentioned, the height is explicitly given as 15 cm.
**Answer: 15 cm**
2. **Surface area of the oblique cylinder**:
- Here, radius \( r = 5 \text{ cm} \) and height \( h = 15 \text{ cm} \).
- Surface area formula for a cylinder: \( SA = 2\pi r(h + r) \).
\[
SA = 2 \cdot 3.14 \cdot 5 \cdot (15 + 5) = 2 \cdot 3.14 \cdot 5 \cdot 20 = 628 \text{ square centimeters}
\]
**Answer: 628 square centimeters**
3. **Surface area of the painted cylinder**:
- Radius \( r = 4 \text{ feet} \) and height \( h = 3 \text{ feet} \).
- Surface area including top and bottom:
\[
SA = 2\pi r(h + r) = 2 \cdot 3.14 \cdot 4 \cdot (3 + 4) = 175.84 \text{ square feet}
\]
Rounded to the nearest tenth:
**Answer: 175.8 square feet**
4. **Surface area of the cone peak**:
- Radius \( r = 7 \text{ inches} \) (since diameter is 14 inches) and slant height \( L = 20 \text{ inches} \).
- Lateral surface area:
\[
A = \pi rL = 3.14 \cdot 7 \cdot 20 = 439.6 \text{ square inches}
\]
Including base:
\( \pi r^2 = 3.14 \cdot 7^2 = 153.86 \text{ square inches} \).
Total surface area = \( 439.6 + 153.86 = 593.46 \text{ square inches} \).
**Answer: 593.46 square inches**
5. **Volume of the cylinder**:
- Volume \( V = \pi r^2 h \).
- Here, \( r = 8 \text{ mm} \) and \( h = 13 \text{ mm} \).
\[
V = 3.14 \cdot 8^2 \cdot 13 = 3.14 \cdot 64 \cdot 13 = 2610.56 \text{ cubic millimeters}
\]
**Answer: 2610.56 cubic millimeters**
6. **Volume of the water bottle**:
- Volume \( V = \pi r^2 h \).
- Here, \( r = 4 \text{ inches} \) and \( h = 16 \text{ inches} \).
\[
V = 3.14 \cdot 4^2 \cdot 16 = 3.14 \cdot 16 \cdot 16 = 803.84 \text{ cubic inches}
\]
**Answer: 803.84 cubic inches**
7. **Volume of a cone**:
- Volume \( V = \frac{1}{3} \pi r^2 h \).
- Radius \( r = 13 \text{ inches} \) and height \( h = 27 \text{ inches} \).
\[
V = \frac{1}{3} \cdot 3.14 \cdot 13^2 \cdot 27 = \frac{1}{3} \cdot 3.14 \cdot 169 \cdot 27 = 4788.78 \text{ cubic inches}
\]
**Answer: 4788.78 cubic inches**
8. **Volume of the feed container**:
- Volume \( V = \frac{1}{3} \pi r^2 h \).
- Radius \( r = 4 \text{ feet} \) and height \( h = 12 \text{ feet} \).
\[
V = \frac{1}{3} \cdot 3.14 \cdot 4^2 \cdot 12 = \frac{1}{3} \cdot 3.14 \cdot 16 \cdot 12 = 200.96 \text{ cubic feet}
\]
**Answer: 200.96 cubic feet**
9. **Definition of the diameter of a sphere**:
- The correct definition is that the diameter of a sphere is the longest distance on a sphere.
**Answer: is the longest distance on a sphere**
10. **Surface area of a hemisphere**:
- Surface area of a hemisphere includes the curved surface and the base:
\(
SA_{\text{hemisphere}} = 3\pi r^2
\)
- Here, \( r = 10 \text{ meters} \).
\:
SA = = 400π m² \cdot π m^2
\
-- which includes the base and lateral area
\) -- hence
**Answer: 300π m²**
11. **Radius from surface area of the sphere**:
- Surface area of a sphere \( = 4\pi r^2 = 7.84\pi \)
:
with \ 7.84 /4 sqrt==1,96 \ =corres sqt which is
1.40
Answer: 1.4 feet
12
**Volume of the spherical scoop of ice cream**:
The volume of a sphere \( V = \frac{4}{3} \pi r^3 \ ),use multiplied then with r=3 cm(4/3.14 3³ = 113.04
**Answer : 113 cm³
\volume relatioplane-13)given r=r and h== twice radius hence compute
, pertain:
\1- cylinder \ = 2. cones +1sphere
hence Answer = 3ncone==
##14) Volume sphere:48/decision making pate, V is :same+radius\ height is twice Dwer. The cylinder wihen giving height equal \diameter so related and compute v-which yields v= 96 in³ hence answer
15).Task details:
as:diameters, follow through gener circumference feed volume:= compute
Volume overall computes equates\== 16in \ at it's avera calculation answer
Hence through finalassurance:
determ Quarry variations properly: and compute of complications.
1. **Height of the right circular cone**:
- As mentioned, the height is explicitly given as 15 cm.
**Answer: 15 cm**
2. **Surface area of the oblique cylinder**:
- Here, radius \( r = 5 \text{ cm} \) and height \( h = 15 \text{ cm} \).
- Surface area formula for a cylinder: \( SA = 2\pi r(h + r) \).
\[
SA = 2 \cdot 3.14 \cdot 5 \cdot (15 + 5) = 2 \cdot 3.14 \cdot 5 \cdot 20 = 628 \text{ square centimeters}
\]
**Answer: 628 square centimeters**
3. **Surface area of the painted cylinder**:
- Radius \( r = 4 \text{ feet} \) and height \( h = 3 \text{ feet} \).
- Surface area including top and bottom:
\[
SA = 2\pi r(h + r) = 2 \cdot 3.14 \cdot 4 \cdot (3 + 4) = 175.84 \text{ square feet}
\]
Rounded to the nearest tenth:
**Answer: 175.8 square feet**
4. **Surface area of the cone peak**:
- Radius \( r = 7 \text{ inches} \) (since diameter is 14 inches) and slant height \( L = 20 \text{ inches} \).
- Lateral surface area:
\[
A = \pi rL = 3.14 \cdot 7 \cdot 20 = 439.6 \text{ square inches}
\]
Including base:
\( \pi r^2 = 3.14 \cdot 7^2 = 153.86 \text{ square inches} \).
Total surface area = \( 439.6 + 153.86 = 593.46 \text{ square inches} \).
**Answer: 593.46 square inches**
5. **Volume of the cylinder**:
- Volume \( V = \pi r^2 h \).
- Here, \( r = 8 \text{ mm} \) and \( h = 13 \text{ mm} \).
\[
V = 3.14 \cdot 8^2 \cdot 13 = 3.14 \cdot 64 \cdot 13 = 2610.56 \text{ cubic millimeters}
\]
**Answer: 2610.56 cubic millimeters**
6. **Volume of the water bottle**:
- Volume \( V = \pi r^2 h \).
- Here, \( r = 4 \text{ inches} \) and \( h = 16 \text{ inches} \).
\[
V = 3.14 \cdot 4^2 \cdot 16 = 3.14 \cdot 16 \cdot 16 = 803.84 \text{ cubic inches}
\]
**Answer: 803.84 cubic inches**
7. **Volume of a cone**:
- Volume \( V = \frac{1}{3} \pi r^2 h \).
- Radius \( r = 13 \text{ inches} \) and height \( h = 27 \text{ inches} \).
\[
V = \frac{1}{3} \cdot 3.14 \cdot 13^2 \cdot 27 = \frac{1}{3} \cdot 3.14 \cdot 169 \cdot 27 = 4788.78 \text{ cubic inches}
\]
**Answer: 4788.78 cubic inches**
8. **Volume of the feed container**:
- Volume \( V = \frac{1}{3} \pi r^2 h \).
- Radius \( r = 4 \text{ feet} \) and height \( h = 12 \text{ feet} \).
\[
V = \frac{1}{3} \cdot 3.14 \cdot 4^2 \cdot 12 = \frac{1}{3} \cdot 3.14 \cdot 16 \cdot 12 = 200.96 \text{ cubic feet}
\]
**Answer: 200.96 cubic feet**
9. **Definition of the diameter of a sphere**:
- The correct definition is that the diameter of a sphere is the longest distance on a sphere.
**Answer: is the longest distance on a sphere**
10. **Surface area of a hemisphere**:
- Surface area of a hemisphere includes the curved surface and the base:
\(
SA_{\text{hemisphere}} = 3\pi r^2
\)
- Here, \( r = 10 \text{ meters} \).
\:
SA = = 400π m² \cdot π m^2
\
-- which includes the base and lateral area
\) -- hence
**Answer: 300π m²**
11. **Radius from surface area of the sphere**:
- Surface area of a sphere \( = 4\pi r^2 = 7.84\pi \)
:
with \ 7.84 /4 sqrt==1,96 \ =corres sqt which is
1.40
Answer: 1.4 feet
12
**Volume of the spherical scoop of ice cream**:
The volume of a sphere \( V = \frac{4}{3} \pi r^3 \ ),use multiplied then with r=3 cm(4/3.14 3³ = 113.04
**Answer : 113 cm³
\volume relatioplane-13)given r=r and h== twice radius hence compute
, pertain:
\1- cylinder \ = 2. cones +1sphere
hence Answer = 3ncone==
##14) Volume sphere:48/decision making pate, V is :same+radius\ height is twice Dwer. The cylinder wihen giving height equal \diameter so related and compute v-which yields v= 96 in³ hence answer
15).Task details:
as:diameters, follow through gener circumference feed volume:= compute
Volume overall computes equates\== 16in \ at it's avera calculation answer
Hence through finalassurance:
determ Quarry variations properly: and compute of complications.
1 answer