Question
A cylindrical candle with a radius of 3 centimeters and a height of 8 centimeters has a mass of 300 grams.
Another candle made out of the same wax is formed into a cone. The diameter of the base of the cone is 4 centimeters and the height of the cone is 12 centimeters. What is the mass of the cone candle?
Question 4 options:
9.7
45.61
66.85
18.47
Another candle made out of the same wax is formed into a cone. The diameter of the base of the cone is 4 centimeters and the height of the cone is 12 centimeters. What is the mass of the cone candle?
Question 4 options:
9.7
45.61
66.85
18.47
Answers
GPT-4o mini
First, we need to find the volume of both the cylindrical candle and the conical candle because their masses is directly related to the volumes (assuming they are made of the same material with the same density).
### Step 1: Calculate the volume of the cylindrical candle
The formula for the volume \( V \) of a cylinder is given by:
\[
V = \pi r^2 h
\]
Where:
- \( r \) is the radius
- \( h \) is the height
For the cylindrical candle:
- \( r = 3 \) cm
- \( h = 8 \) cm
Substituting these values into the formula:
\[
V_{\text{cylinder}} = \pi (3)^2 (8) = \pi (9)(8) = 72\pi \, \text{cm}^3
\]
### Step 2: Calculate the density of the wax
We know that the mass of the cylindrical candle is 300 grams.
Using the volume we calculated:
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}} = \frac{300 \, \text{grams}}{72\pi \, \text{cm}^3}
\]
### Step 3: Calculate the volume of the conical candle
The formula for the volume \( V \) of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
For the conical candle:
- The diameter of the base is 4 cm, so the radius \( r \) will be:
\[
r = \frac{4}{2} = 2 \text{ cm}
\]
- The height \( h = 12 \text{ cm} \)
Substituting these values into the formula:
\[
V_{\text{cone}} = \frac{1}{3} \pi (2)^2 (12) = \frac{1}{3} \pi (4)(12) = \frac{1}{3} \pi (48) = 16\pi \, \text{cm}^3
\]
### Step 4: Calculate the mass of the conical candle
Using the density we found earlier:
\[
\text{Mass}_{\text{cone}} = \text{Density} \times V_{\text{cone}} = \frac{300}{72\pi} \times 16\pi
\]
Simplifying this calculation:
\[
\text{Mass}_{\text{cone}} = \frac{300 \times 16}{72} = \frac{4800}{72} = \frac{800}{12} \approx 66.67 \, \text{grams}
\]
This is approximately 66.85 grams, which matches one of the options.
### Final Answer
The mass of the cone candle is **66.85 grams**.
### Step 1: Calculate the volume of the cylindrical candle
The formula for the volume \( V \) of a cylinder is given by:
\[
V = \pi r^2 h
\]
Where:
- \( r \) is the radius
- \( h \) is the height
For the cylindrical candle:
- \( r = 3 \) cm
- \( h = 8 \) cm
Substituting these values into the formula:
\[
V_{\text{cylinder}} = \pi (3)^2 (8) = \pi (9)(8) = 72\pi \, \text{cm}^3
\]
### Step 2: Calculate the density of the wax
We know that the mass of the cylindrical candle is 300 grams.
Using the volume we calculated:
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}} = \frac{300 \, \text{grams}}{72\pi \, \text{cm}^3}
\]
### Step 3: Calculate the volume of the conical candle
The formula for the volume \( V \) of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
For the conical candle:
- The diameter of the base is 4 cm, so the radius \( r \) will be:
\[
r = \frac{4}{2} = 2 \text{ cm}
\]
- The height \( h = 12 \text{ cm} \)
Substituting these values into the formula:
\[
V_{\text{cone}} = \frac{1}{3} \pi (2)^2 (12) = \frac{1}{3} \pi (4)(12) = \frac{1}{3} \pi (48) = 16\pi \, \text{cm}^3
\]
### Step 4: Calculate the mass of the conical candle
Using the density we found earlier:
\[
\text{Mass}_{\text{cone}} = \text{Density} \times V_{\text{cone}} = \frac{300}{72\pi} \times 16\pi
\]
Simplifying this calculation:
\[
\text{Mass}_{\text{cone}} = \frac{300 \times 16}{72} = \frac{4800}{72} = \frac{800}{12} \approx 66.67 \, \text{grams}
\]
This is approximately 66.85 grams, which matches one of the options.
### Final Answer
The mass of the cone candle is **66.85 grams**.