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You work for a perfume company and are being asked to design two new perfume bottles to go with a line of perfume for teen girl...Question
You work for a perfume company and are being asked to design two new perfume bottles to go with a line of perfume for teen girls. The bottle they have now is a cylinder and has a base radius of 2 cm and a height of 9 cm. They want each of the new bottles to hold the same about of perfume as the original but for them to each be a different shape. One should be a cone, and one a sphere.
You are in charge of designing the bottles, what dimensions would you choose for the sphere and cone shaped bottles? Explain how you know that each bottle would hold the same amount of liquid as the original bottle.
Josiah tried the problem above and decides that the best way to make them all the same volume is the keep them all with a radius of 2 and a height of 9. That way, each bottle is the same size and will hold the same volume.
Do you agree with Josiah, why or why not? Use mathematical reasoning to support your answer.
You are in charge of designing the bottles, what dimensions would you choose for the sphere and cone shaped bottles? Explain how you know that each bottle would hold the same amount of liquid as the original bottle.
Josiah tried the problem above and decides that the best way to make them all the same volume is the keep them all with a radius of 2 and a height of 9. That way, each bottle is the same size and will hold the same volume.
Do you agree with Josiah, why or why not? Use mathematical reasoning to support your answer.
Answers
GPT-4o mini
To design two new perfume bottles (one cone-shaped and one sphere-shaped) that hold the same volume as the existing cylindrical bottle, we first need to determine the volume of the original cylinder.
### Step 1: Calculate the Volume of the Cylinder
The formula for the volume \( V \) of a cylinder is given by:
\[
V = \pi r^2 h
\]
where:
- \( r \) is the radius of the base
- \( h \) is the height
For the original bottle:
- Base radius \( r = 2 \) cm
- Height \( h = 9 \) cm
Calculating the volume:
\[
V = \pi (2)^2 (9) = \pi (4)(9) = 36\pi \text{ cm}^3
\]
### Step 2: Cone-Shaped Bottle
The volume \( V \) of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
To find a cone with the same volume \( 36\pi \text{ cm}^3 \), we can set up the equation:
\[
\frac{1}{3} \pi r^2 h = 36\pi
\]
Dividing both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[
\frac{1}{3} r^2 h = 36
\]
Multiplying both sides by 3:
\[
r^2 h = 108
\]
Now, we can choose a suitable height \( h \) for the cone, and calculate the associated radius \( r \) that will make the volume equal to the cylinder's volume.
**Let’s choose a height of 6 cm:**
Then,
\[
r^2(6) = 108 \implies r^2 = 18 \implies r = \sqrt{18} \approx 4.24 \text{ cm}
\]
So, one option for the cone is:
- **Height**: 6 cm
- **Radius**: \( \sqrt{18} \approx 4.24 \) cm
### Step 3: Sphere-Shaped Bottle
The volume \( V \) of a sphere is given by:
\[
V = \frac{4}{3} \pi r^3
\]
We want this volume to be \( 36\pi \text{ cm}^3 \):
\[
\frac{4}{3} \pi r^3 = 36\pi
\]
Dividing by \( \pi \):
\[
\frac{4}{3} r^3 = 36
\]
Multiplying both sides by \( \frac{3}{4} \):
\[
r^3 = 27
\]
Taking the cube root:
\[
r = 3 \text{ cm}
\]
So, for the sphere, the dimensions would be:
- **Radius**: 3 cm
### Summary of Dimensions
- **Cone**: Radius \( \sqrt{18} \approx 4.24 \) cm, Height 6 cm
- **Sphere**: Radius 3 cm
### Agreement with Josiah's Approach
Now, addressing Josiah’s approach, he suggested keeping the radius at 2 cm and the height at 9 cm for all shapes. This would keep all bottles the same volume as the cylinder. However, while this would create identical volumes, it defeats the purpose of designing new shapes.
Let’s calculate the volume when keeping radius 2 cm and height 9 cm:
1. **Cone Volume**:
\[
V = \frac{1}{3} \pi (2)^2 (9) = \frac{1}{3} \pi (4)(9) = 12\pi \text{ cm}^3 \quad (\text{not the same as the cylinder})
\]
2. **Sphere Volume**:
\[
V = \frac{4}{3} \pi (2)^3 = \frac{4}{3} \pi (8) = \frac{32}{3}\pi \text{ cm}^3 \quad (\text{not the same as the cylinder})
\]
In conclusion, Josiah's suggestion does not hold because the volumes do not match the cylinder's volume, but my designs using the appropriate formulas do achieve the desired result of equal volume while varying the shapes. Therefore, I do not agree with Josiah's approach.
### Step 1: Calculate the Volume of the Cylinder
The formula for the volume \( V \) of a cylinder is given by:
\[
V = \pi r^2 h
\]
where:
- \( r \) is the radius of the base
- \( h \) is the height
For the original bottle:
- Base radius \( r = 2 \) cm
- Height \( h = 9 \) cm
Calculating the volume:
\[
V = \pi (2)^2 (9) = \pi (4)(9) = 36\pi \text{ cm}^3
\]
### Step 2: Cone-Shaped Bottle
The volume \( V \) of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
To find a cone with the same volume \( 36\pi \text{ cm}^3 \), we can set up the equation:
\[
\frac{1}{3} \pi r^2 h = 36\pi
\]
Dividing both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[
\frac{1}{3} r^2 h = 36
\]
Multiplying both sides by 3:
\[
r^2 h = 108
\]
Now, we can choose a suitable height \( h \) for the cone, and calculate the associated radius \( r \) that will make the volume equal to the cylinder's volume.
**Let’s choose a height of 6 cm:**
Then,
\[
r^2(6) = 108 \implies r^2 = 18 \implies r = \sqrt{18} \approx 4.24 \text{ cm}
\]
So, one option for the cone is:
- **Height**: 6 cm
- **Radius**: \( \sqrt{18} \approx 4.24 \) cm
### Step 3: Sphere-Shaped Bottle
The volume \( V \) of a sphere is given by:
\[
V = \frac{4}{3} \pi r^3
\]
We want this volume to be \( 36\pi \text{ cm}^3 \):
\[
\frac{4}{3} \pi r^3 = 36\pi
\]
Dividing by \( \pi \):
\[
\frac{4}{3} r^3 = 36
\]
Multiplying both sides by \( \frac{3}{4} \):
\[
r^3 = 27
\]
Taking the cube root:
\[
r = 3 \text{ cm}
\]
So, for the sphere, the dimensions would be:
- **Radius**: 3 cm
### Summary of Dimensions
- **Cone**: Radius \( \sqrt{18} \approx 4.24 \) cm, Height 6 cm
- **Sphere**: Radius 3 cm
### Agreement with Josiah's Approach
Now, addressing Josiah’s approach, he suggested keeping the radius at 2 cm and the height at 9 cm for all shapes. This would keep all bottles the same volume as the cylinder. However, while this would create identical volumes, it defeats the purpose of designing new shapes.
Let’s calculate the volume when keeping radius 2 cm and height 9 cm:
1. **Cone Volume**:
\[
V = \frac{1}{3} \pi (2)^2 (9) = \frac{1}{3} \pi (4)(9) = 12\pi \text{ cm}^3 \quad (\text{not the same as the cylinder})
\]
2. **Sphere Volume**:
\[
V = \frac{4}{3} \pi (2)^3 = \frac{4}{3} \pi (8) = \frac{32}{3}\pi \text{ cm}^3 \quad (\text{not the same as the cylinder})
\]
In conclusion, Josiah's suggestion does not hold because the volumes do not match the cylinder's volume, but my designs using the appropriate formulas do achieve the desired result of equal volume while varying the shapes. Therefore, I do not agree with Josiah's approach.