Question
To find the volume of the cone in the picture where S= 13cm and R= 5cm:
Step 1 _____________
Step 2 _____________
Step 3 _____________
find h by using the area of a triangle= (b-h)/2
find h by using the Pythagorean theorem r^2 + h^2 = s^2
since h=6, use the formula V= 1/3pir^2h to find the volume
since h=12, use the formula V= 1/3pir^2h to find the volume
the volume is approximately 314.2cm^3
the volume is approximately 38.04cm^3
Step 1 _____________
Step 2 _____________
Step 3 _____________
find h by using the area of a triangle= (b-h)/2
find h by using the Pythagorean theorem r^2 + h^2 = s^2
since h=6, use the formula V= 1/3pir^2h to find the volume
since h=12, use the formula V= 1/3pir^2h to find the volume
the volume is approximately 314.2cm^3
the volume is approximately 38.04cm^3
Answers
GPT-4o mini
To find the volume of the cone given slant height \( S = 13 \, \text{cm} \) and radius \( R = 5 \, \text{cm} \), we can follow these steps:
### Step 1: Find the height \( h \) of the cone using the Pythagorean theorem
Since we are given the radius (base) \( R \) and the slant height \( S \), we can use the Pythagorean theorem:
\[
h^2 + R^2 = S^2
\]
Substituting the known values, we get:
\[
h^2 + 5^2 = 13^2
\]
\[
h^2 + 25 = 169
\]
\[
h^2 = 169 - 25
\]
\[
h^2 = 144
\]
\[
h = \sqrt{144} = 12 \, \text{cm}
\]
### Step 2: Calculate the volume \( V \) of the cone using the volume formula
The volume \( V \) of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
Substituting the values of \( r \) and \( h \) into the formula:
\[
V = \frac{1}{3} \pi (5^2)(12)
\]
\[
V = \frac{1}{3} \pi (25)(12)
\]
\[
V = \frac{1}{3} \pi (300)
\]
\[
V = 100\pi \approx 314.16 \, \text{cm}^3
\]
### Step 3: State the final volume of the cone
Thus, the volume of the cone is approximately:
\[
V \approx 314.16 \, \text{cm}^3
\]
### Summary
- Step 1: Use Pythagorean theorem to find height \( h \).
- Step 2: Apply the volume formula for cones to find \( V \).
- Step 3: State the volume, approximately \( 314.2 \, \text{cm}^3 \).
Note: There was an incorrect mention of using \( h = 6 \). The correct height calculated is \( h = 12 \, \text{cm} \). The calculations for volume around \( 38.04 \, \text{cm}^3 \) are not relevant in this particular context with the given measurements.
### Step 1: Find the height \( h \) of the cone using the Pythagorean theorem
Since we are given the radius (base) \( R \) and the slant height \( S \), we can use the Pythagorean theorem:
\[
h^2 + R^2 = S^2
\]
Substituting the known values, we get:
\[
h^2 + 5^2 = 13^2
\]
\[
h^2 + 25 = 169
\]
\[
h^2 = 169 - 25
\]
\[
h^2 = 144
\]
\[
h = \sqrt{144} = 12 \, \text{cm}
\]
### Step 2: Calculate the volume \( V \) of the cone using the volume formula
The volume \( V \) of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
Substituting the values of \( r \) and \( h \) into the formula:
\[
V = \frac{1}{3} \pi (5^2)(12)
\]
\[
V = \frac{1}{3} \pi (25)(12)
\]
\[
V = \frac{1}{3} \pi (300)
\]
\[
V = 100\pi \approx 314.16 \, \text{cm}^3
\]
### Step 3: State the final volume of the cone
Thus, the volume of the cone is approximately:
\[
V \approx 314.16 \, \text{cm}^3
\]
### Summary
- Step 1: Use Pythagorean theorem to find height \( h \).
- Step 2: Apply the volume formula for cones to find \( V \).
- Step 3: State the volume, approximately \( 314.2 \, \text{cm}^3 \).
Note: There was an incorrect mention of using \( h = 6 \). The correct height calculated is \( h = 12 \, \text{cm} \). The calculations for volume around \( 38.04 \, \text{cm}^3 \) are not relevant in this particular context with the given measurements.