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.
You may explain your work in words or do your work in a jamboard and input the link to it here. You may also do your work on paper and submit that.
You will get 2 points for the dimensions of each bottle and 5 points for your explanation as to how you know they will hold the same volume. You will get 1 point for using complete sentences and good grammar.
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.
You may explain your work in words or do your work in a jamboard and input the link to it here. You may also do your work on paper and submit that.
You will get 2 points for the dimensions of each bottle and 5 points for your explanation as to how you know they will hold the same volume. You will get 1 point for using complete sentences and good grammar.
Answers
GPT-4o mini
To ensure that the new perfume bottles hold the same volume as the original cylindrical bottle, we need to first find the volume of the cylinder and then use that volume to determine the dimensions of the cone and the sphere.
### 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 of the cylinder.
Given:
- Base radius \( r = 2 \) cm
- Height \( h = 9 \) cm
Substituting the values in:
\[
V = \pi (2)^2 (9) = \pi (4)(9) = 36\pi \text{ cm}^3
\]
### Step 2: Dimensions for the Cone Bottle
The formula for the volume \( V \) of a cone is:
\[
V = \frac{1}{3} \pi r^2 h
\]
We know the volume must be \( 36\pi \text{ cm}^3 \).
Setting the volume of the cone equal to the cylinder’s volume:
\[
\frac{1}{3} \pi r^2 h = 36\pi
\]
Dividing both sides by \( \pi \):
\[
\frac{1}{3} r^2 h = 36
\]
Multiplying by 3:
\[
r^2 h = 108
\]
We can select a radius \( r \) and solve for height \( h \). Let’s choose \( r = 3 \) cm.
Now we find \( h \):
\[
(3)^2 h = 108
\]
\[
9h = 108
\]
\[
h = \frac{108}{9} = 12 \text{ cm}
\]
So, the dimensions for the cone-shaped bottle will be:
- Radius \( r = 3 \) cm
- Height \( h = 12 \) cm
### Step 3: Dimensions for the Sphere Bottle
The formula for the volume \( V \) of a sphere is:
\[
V = \frac{4}{3} \pi r^3
\]
Again, we set this equal to the volume of the cylinder:
\[
\frac{4}{3} \pi r^3 = 36\pi
\]
Dividing both sides by \( \pi \):
\[
\frac{4}{3} r^3 = 36
\]
Multiplying both sides by \( \frac{3}{4} \):
\[
r^3 = 36 \cdot \frac{3}{4}
\]
\[
r^3 = 27
\]
Taking the cube root:
\[
r = 3 \text{ cm}
\]
So, the dimensions for the sphere-shaped bottle will be:
- Radius \( r = 3 \) cm
### Summary of the Dimensions:
- **Cone:** Radius = 3 cm, Height = 12 cm
- **Sphere:** Radius = 3 cm
### Explanation of Volume Equivalence
Both the cone and sphere have been designed to hold the same volume as the original cylindrical bottle, which is \( 36\pi \text{ cm}^3 \).
- For the cone, we validated our design by ensuring that \( \frac{1}{3} \pi (3^2)(12) = 36\pi \).
- For the sphere, the volume calculation \( \frac{4}{3} \pi (3^3) = 36\pi \) confirms that it meets the original volume requirement.
In conclusion, each new bottle design has been calculated accurately to ensure it retains the same capacity as the original cylindrical bottle, fulfilling the design requirements for the perfume line targeted at teen girls.
### 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 of the cylinder.
Given:
- Base radius \( r = 2 \) cm
- Height \( h = 9 \) cm
Substituting the values in:
\[
V = \pi (2)^2 (9) = \pi (4)(9) = 36\pi \text{ cm}^3
\]
### Step 2: Dimensions for the Cone Bottle
The formula for the volume \( V \) of a cone is:
\[
V = \frac{1}{3} \pi r^2 h
\]
We know the volume must be \( 36\pi \text{ cm}^3 \).
Setting the volume of the cone equal to the cylinder’s volume:
\[
\frac{1}{3} \pi r^2 h = 36\pi
\]
Dividing both sides by \( \pi \):
\[
\frac{1}{3} r^2 h = 36
\]
Multiplying by 3:
\[
r^2 h = 108
\]
We can select a radius \( r \) and solve for height \( h \). Let’s choose \( r = 3 \) cm.
Now we find \( h \):
\[
(3)^2 h = 108
\]
\[
9h = 108
\]
\[
h = \frac{108}{9} = 12 \text{ cm}
\]
So, the dimensions for the cone-shaped bottle will be:
- Radius \( r = 3 \) cm
- Height \( h = 12 \) cm
### Step 3: Dimensions for the Sphere Bottle
The formula for the volume \( V \) of a sphere is:
\[
V = \frac{4}{3} \pi r^3
\]
Again, we set this equal to the volume of the cylinder:
\[
\frac{4}{3} \pi r^3 = 36\pi
\]
Dividing both sides by \( \pi \):
\[
\frac{4}{3} r^3 = 36
\]
Multiplying both sides by \( \frac{3}{4} \):
\[
r^3 = 36 \cdot \frac{3}{4}
\]
\[
r^3 = 27
\]
Taking the cube root:
\[
r = 3 \text{ cm}
\]
So, the dimensions for the sphere-shaped bottle will be:
- Radius \( r = 3 \) cm
### Summary of the Dimensions:
- **Cone:** Radius = 3 cm, Height = 12 cm
- **Sphere:** Radius = 3 cm
### Explanation of Volume Equivalence
Both the cone and sphere have been designed to hold the same volume as the original cylindrical bottle, which is \( 36\pi \text{ cm}^3 \).
- For the cone, we validated our design by ensuring that \( \frac{1}{3} \pi (3^2)(12) = 36\pi \).
- For the sphere, the volume calculation \( \frac{4}{3} \pi (3^3) = 36\pi \) confirms that it meets the original volume requirement.
In conclusion, each new bottle design has been calculated accurately to ensure it retains the same capacity as the original cylindrical bottle, fulfilling the design requirements for the perfume line targeted at teen girls.