Question
1. What is the length of the radius of a sphere with a diameter of 12 cm?(1 point)
cm
2. Find the surface area of a sphere with a diameter of 40 inches. Leave your answer in terms of pi.(1 point)
The surface area is in.^2
3. Find the surface area of a sphere with a radius of 3.6 ft. Use 3.14 for pi. Round your answer to the nearest hundredth, if necessary.(1 point)
The surface area is ft.^2
4. A cube box is 20 cm×20 cm×20 cm
20
cm
×
20
cm
×
20
cm
. What is the surface area of the largest size sphere that can fit in this box? Leave your answer in terms of pi.(1 point)
cm^2
5. A rectangular box is 12 in. × 14 in. × 18 in.
12
in
.
×
14
in
.
×
18
in
.
What is the surface area of the largest size sphere that could fit in this box? Leave your answer in terms of pi.(1 point)
in^2
cm
2. Find the surface area of a sphere with a diameter of 40 inches. Leave your answer in terms of pi.(1 point)
The surface area is in.^2
3. Find the surface area of a sphere with a radius of 3.6 ft. Use 3.14 for pi. Round your answer to the nearest hundredth, if necessary.(1 point)
The surface area is ft.^2
4. A cube box is 20 cm×20 cm×20 cm
20
cm
×
20
cm
×
20
cm
. What is the surface area of the largest size sphere that can fit in this box? Leave your answer in terms of pi.(1 point)
cm^2
5. A rectangular box is 12 in. × 14 in. × 18 in.
12
in
.
×
14
in
.
×
18
in
.
What is the surface area of the largest size sphere that could fit in this box? Leave your answer in terms of pi.(1 point)
in^2
Answers
GPT-4o mini
Here are the answers to each of your questions:
1. **What is the length of the radius of a sphere with a diameter of 12 cm?**
- The radius is half of the diameter.
- Radius = Diameter / 2 = 12 cm / 2 = **6 cm**
2. **Find the surface area of a sphere with a diameter of 40 inches. Leave your answer in terms of pi.**
- The formula for the surface area of a sphere is \( S = 4\pi r^2 \).
- First, find the radius: \( r = \text{Diameter} / 2 = 40 \text{ in} / 2 = 20 \text{ in} \).
- Surface area \( S = 4\pi (20 \text{ in})^2 = 4\pi (400) = 1600\pi \).
- The surface area is **1600π in²**.
3. **Find the surface area of a sphere with a radius of 3.6 ft. Use 3.14 for pi. Round your answer to the nearest hundredth, if necessary.**
- Using the formula \( S = 4\pi r^2 \):
- Surface area \( S = 4 \cdot 3.14 \cdot (3.6 \text{ ft})^2 = 4 \cdot 3.14 \cdot 12.96 \).
- Calculating that: \( 4 \cdot 3.14 = 12.56 \).
- Then \( 12.56 \cdot 12.96 \approx 162.74 \) ft².
- The surface area is **162.74 ft²**.
4. **What is the surface area of the largest size sphere that can fit in a box measuring 20 cm × 20 cm × 20 cm? Leave your answer in terms of pi.**
- The largest sphere that can fit inside the cube will have a diameter equal to the side length of the cube.
- Diameter = 20 cm ⇒ Radius = 10 cm.
- Using \( S = 4\pi r^2 \):
- Surface area \( S = 4\pi (10 \text{ cm})^2 = 4\pi (100) = 400\pi \).
- The surface area is **400π cm²**.
5. **What is the surface area of the largest size sphere that could fit in a box measuring 12 in. × 14 in. × 18 in.? Leave your answer in terms of pi.**
- The largest sphere that can fit in the rectangular box will have a diameter equal to the smallest dimension of the box.
- The smallest dimension is 12 in. ⇒ Radius = 6 in.
- Using the formula \( S = 4\pi r^2 \):
- Surface area \( S = 4\pi (6 \text{ in})^2 = 4\pi (36) = 144\pi \).
- The surface area is **144π in²**.
Here is a summary of the answers:
1. **6 cm**
2. **1600π in²**
3. **162.74 ft²**
4. **400π cm²**
5. **144π in²**
1. **What is the length of the radius of a sphere with a diameter of 12 cm?**
- The radius is half of the diameter.
- Radius = Diameter / 2 = 12 cm / 2 = **6 cm**
2. **Find the surface area of a sphere with a diameter of 40 inches. Leave your answer in terms of pi.**
- The formula for the surface area of a sphere is \( S = 4\pi r^2 \).
- First, find the radius: \( r = \text{Diameter} / 2 = 40 \text{ in} / 2 = 20 \text{ in} \).
- Surface area \( S = 4\pi (20 \text{ in})^2 = 4\pi (400) = 1600\pi \).
- The surface area is **1600π in²**.
3. **Find the surface area of a sphere with a radius of 3.6 ft. Use 3.14 for pi. Round your answer to the nearest hundredth, if necessary.**
- Using the formula \( S = 4\pi r^2 \):
- Surface area \( S = 4 \cdot 3.14 \cdot (3.6 \text{ ft})^2 = 4 \cdot 3.14 \cdot 12.96 \).
- Calculating that: \( 4 \cdot 3.14 = 12.56 \).
- Then \( 12.56 \cdot 12.96 \approx 162.74 \) ft².
- The surface area is **162.74 ft²**.
4. **What is the surface area of the largest size sphere that can fit in a box measuring 20 cm × 20 cm × 20 cm? Leave your answer in terms of pi.**
- The largest sphere that can fit inside the cube will have a diameter equal to the side length of the cube.
- Diameter = 20 cm ⇒ Radius = 10 cm.
- Using \( S = 4\pi r^2 \):
- Surface area \( S = 4\pi (10 \text{ cm})^2 = 4\pi (100) = 400\pi \).
- The surface area is **400π cm²**.
5. **What is the surface area of the largest size sphere that could fit in a box measuring 12 in. × 14 in. × 18 in.? Leave your answer in terms of pi.**
- The largest sphere that can fit in the rectangular box will have a diameter equal to the smallest dimension of the box.
- The smallest dimension is 12 in. ⇒ Radius = 6 in.
- Using the formula \( S = 4\pi r^2 \):
- Surface area \( S = 4\pi (6 \text{ in})^2 = 4\pi (36) = 144\pi \).
- The surface area is **144π in²**.
Here is a summary of the answers:
1. **6 cm**
2. **1600π in²**
3. **162.74 ft²**
4. **400π cm²**
5. **144π in²**
1. Which of the following correctly describes a sphere?(1 point)
Responses
A sphere has a vertex.
A sphere has a vertex.
A sphere has a face.
A sphere has a face.
A sphere has a center.
A sphere has a center.
A sphere has an edge.
2. Which of the following is true about the diameter of a sphere?(1 point)
Responses
The diameter of a sphere passes through the center of the circle.
The diameter of a sphere passes through the center of the circle.
The diameter is not a chord.
The diameter is not a chord.
The diameter is 2.5 times the length of the radius.
The diameter is 2.5 times the length of the radius.
The diameter is the shortest length on a sphere.
3. Use the formula for the surface area of a sphere to find the surface area of a basketball that has a diameter of 9.2 inches. Use 3.14 pi and round your answer to the nearest hundredth, if necessary.(1 point)
Responses
66.44 in.^2
1,063.08 in.^2
84.64 in.^2
265.77 in.^2
Find the surface area of a sphere with a diameter of 30 inches. Leave your answer in terms of pi.(1 point)
Responses
3,600π in.^2
2,826π in.^2
225π in.^2
900π in.^2
5. Marcus is making spherical soaps to sell in his online store. The surface area of a soap is 63.585 in.^2
63.585
in
.
2
and he wants to package them into a cube box so that it fits snugly. Using 3.14 as the value of pi, what should be the dimensions of the cube box?(1 point)
Responses
4.5 in. × 4.5 in. × 4.5 in.
4.5
in.
×
4.5
in.
×
4.5
in
.
4.5 inches times 4.5 inches times 4.5 inches
4 in.× 4 in.× 4 in.
4
in.
×
4
in.
×
4
in
.
4 inches times 4 inches times 4 inches
2.25 in. × 2.25 in. × 2.25 in.
2.25
in.
×
2.25
in.
×
2.25
in
.
2.25 inches times 2.25 inches times 2.25 inches
1.2 in. × 1.2 in. × 1.2 in.
Responses
A sphere has a vertex.
A sphere has a vertex.
A sphere has a face.
A sphere has a face.
A sphere has a center.
A sphere has a center.
A sphere has an edge.
2. Which of the following is true about the diameter of a sphere?(1 point)
Responses
The diameter of a sphere passes through the center of the circle.
The diameter of a sphere passes through the center of the circle.
The diameter is not a chord.
The diameter is not a chord.
The diameter is 2.5 times the length of the radius.
The diameter is 2.5 times the length of the radius.
The diameter is the shortest length on a sphere.
3. Use the formula for the surface area of a sphere to find the surface area of a basketball that has a diameter of 9.2 inches. Use 3.14 pi and round your answer to the nearest hundredth, if necessary.(1 point)
Responses
66.44 in.^2
1,063.08 in.^2
84.64 in.^2
265.77 in.^2
Find the surface area of a sphere with a diameter of 30 inches. Leave your answer in terms of pi.(1 point)
Responses
3,600π in.^2
2,826π in.^2
225π in.^2
900π in.^2
5. Marcus is making spherical soaps to sell in his online store. The surface area of a soap is 63.585 in.^2
63.585
in
.
2
and he wants to package them into a cube box so that it fits snugly. Using 3.14 as the value of pi, what should be the dimensions of the cube box?(1 point)
Responses
4.5 in. × 4.5 in. × 4.5 in.
4.5
in.
×
4.5
in.
×
4.5
in
.
4.5 inches times 4.5 inches times 4.5 inches
4 in.× 4 in.× 4 in.
4
in.
×
4
in.
×
4
in
.
4 inches times 4 inches times 4 inches
2.25 in. × 2.25 in. × 2.25 in.
2.25
in.
×
2.25
in.
×
2.25
in
.
2.25 inches times 2.25 inches times 2.25 inches
1.2 in. × 1.2 in. × 1.2 in.
GPT-4o mini
Here are the answers to your questions:
1. **Which of the following correctly describes a sphere?**
- The correct response is: **A sphere has a center.**
(A sphere does not have a vertex, face, or edge.)
2. **Which of the following is true about the diameter of a sphere?**
- The correct response is: **The diameter of a sphere passes through the center of the circle.**
(The diameter is a special type of chord, it is not the shortest length on a sphere, nor is it 2.5 times the length of the radius; the diameter is exactly 2 times the radius.)
3. **Use the formula for the surface area of a sphere to find the surface area of a basketball that has a diameter of 9.2 inches. Use 3.14 for pi and round your answer to the nearest hundredth, if necessary.**
- First, find the radius:
- Radius = Diameter / 2 = 9.2 in / 2 = 4.6 in.
- Surface area \( S = 4\pi r^2 = 4 \cdot 3.14 \cdot (4.6)^2 \).
- Calculate \( (4.6)^2 = 21.16 \).
- Now \( S = 4 \cdot 3.14 \cdot 21.16 = 4 \cdot 66.47584 = 265.90336 \).
- Rounding to the nearest hundredth gives: **265.90 in².**
- Since the closest option is **265.77 in²**, that is the correct response in this context.
4. **Find the surface area of a sphere with a diameter of 30 inches. Leave your answer in terms of pi.**
- First, find the radius:
- Radius = Diameter / 2 = 30 in / 2 = 15 in.
- Surface area \( S = 4\pi r^2 = 4\pi (15)^2 = 4\pi (225) = 900\pi \).
- The correct response is: **900π in².**
5. **Marcus is making spherical soaps to sell in his online store. The surface area of a soap is 63.585 in² and he wants to package them into a cube box so that it fits snugly. Using 3.14 as the value of pi, what should be the dimensions of the cube box?**
- First, we need to find the radius from the surface area:
- The surface area \( S = 4\pi r^2 \) ⇒ \( 63.585 = 4 \cdot 3.14 \cdot r^2 \).
- Calculate \( 4 \cdot 3.14 = 12.56 \), then:
- \( 63.585 = 12.56 \cdot r^2 \)
- Dividing both sides by 12.56 gives us \( r^2 = 63.585 / 12.56 \approx 5.06 \).
- Taking the square root to find the radius: \( r \approx 2.25 \) in.
- Since the largest sphere that can fit in the cube has a diameter of \( 2 \cdot 2.25 = 4.5 \) in, the dimensions of the cube box should be:
- **4.5 in. × 4.5 in. × 4.5 in.**
Here’s a summary of the answers:
1. **A sphere has a center.**
2. **The diameter of a sphere passes through the center of the circle.**
3. **265.77 in²** (or **265.90 in²** if we round correctly)
4. **900π in².**
5. **4.5 in. × 4.5 in. × 4.5 in.**
1. **Which of the following correctly describes a sphere?**
- The correct response is: **A sphere has a center.**
(A sphere does not have a vertex, face, or edge.)
2. **Which of the following is true about the diameter of a sphere?**
- The correct response is: **The diameter of a sphere passes through the center of the circle.**
(The diameter is a special type of chord, it is not the shortest length on a sphere, nor is it 2.5 times the length of the radius; the diameter is exactly 2 times the radius.)
3. **Use the formula for the surface area of a sphere to find the surface area of a basketball that has a diameter of 9.2 inches. Use 3.14 for pi and round your answer to the nearest hundredth, if necessary.**
- First, find the radius:
- Radius = Diameter / 2 = 9.2 in / 2 = 4.6 in.
- Surface area \( S = 4\pi r^2 = 4 \cdot 3.14 \cdot (4.6)^2 \).
- Calculate \( (4.6)^2 = 21.16 \).
- Now \( S = 4 \cdot 3.14 \cdot 21.16 = 4 \cdot 66.47584 = 265.90336 \).
- Rounding to the nearest hundredth gives: **265.90 in².**
- Since the closest option is **265.77 in²**, that is the correct response in this context.
4. **Find the surface area of a sphere with a diameter of 30 inches. Leave your answer in terms of pi.**
- First, find the radius:
- Radius = Diameter / 2 = 30 in / 2 = 15 in.
- Surface area \( S = 4\pi r^2 = 4\pi (15)^2 = 4\pi (225) = 900\pi \).
- The correct response is: **900π in².**
5. **Marcus is making spherical soaps to sell in his online store. The surface area of a soap is 63.585 in² and he wants to package them into a cube box so that it fits snugly. Using 3.14 as the value of pi, what should be the dimensions of the cube box?**
- First, we need to find the radius from the surface area:
- The surface area \( S = 4\pi r^2 \) ⇒ \( 63.585 = 4 \cdot 3.14 \cdot r^2 \).
- Calculate \( 4 \cdot 3.14 = 12.56 \), then:
- \( 63.585 = 12.56 \cdot r^2 \)
- Dividing both sides by 12.56 gives us \( r^2 = 63.585 / 12.56 \approx 5.06 \).
- Taking the square root to find the radius: \( r \approx 2.25 \) in.
- Since the largest sphere that can fit in the cube has a diameter of \( 2 \cdot 2.25 = 4.5 \) in, the dimensions of the cube box should be:
- **4.5 in. × 4.5 in. × 4.5 in.**
Here’s a summary of the answers:
1. **A sphere has a center.**
2. **The diameter of a sphere passes through the center of the circle.**
3. **265.77 in²** (or **265.90 in²** if we round correctly)
4. **900π in².**
5. **4.5 in. × 4.5 in. × 4.5 in.**
3. for PRACTICE is 162.78
all for the quick check is PERFECT! same with 1 2 4 5 in practice.
all for the quick check is PERFECT! same with 1 2 4 5 in practice.
GPT-4o mini
Great to hear that you found the answers for the quick check to be perfect! Thank you for the update on question 3 in your practice; if the calculated surface area for the basketball is approximately **162.78 in²**, that aligns more closely with your rounding approach.
To clarify, for the basketball with a diameter of 9.2 inches:
1. **Calculate the radius**: \[ \text{Radius} = \frac{9.2}{2} = 4.6 \text{ in}\]
2. **Use the surface area formula**:
\[
S = 4\pi r^2 = 4 \cdot 3.14 \cdot (4.6)^2
\]
\[
(4.6)^2 \approx 21.16
\]
\[
S \approx 4 \cdot 3.14 \cdot 21.16 \approx 4 \cdot 66.4816 \approx 265.93 \text{ in²}
\]
However, if you’re using a different calculation or rounding method that gives **162.78 in²**, I'd recommend double-checking the surface area formula and calculations.
If you need further clarification or assistance on any other topics, feel free to ask!
To clarify, for the basketball with a diameter of 9.2 inches:
1. **Calculate the radius**: \[ \text{Radius} = \frac{9.2}{2} = 4.6 \text{ in}\]
2. **Use the surface area formula**:
\[
S = 4\pi r^2 = 4 \cdot 3.14 \cdot (4.6)^2
\]
\[
(4.6)^2 \approx 21.16
\]
\[
S \approx 4 \cdot 3.14 \cdot 21.16 \approx 4 \cdot 66.4816 \approx 265.93 \text{ in²}
\]
However, if you’re using a different calculation or rounding method that gives **162.78 in²**, I'd recommend double-checking the surface area formula and calculations.
If you need further clarification or assistance on any other topics, feel free to ask!