Question
A cone has a height of h, a radius of 15 inches, and a slant height of 17 inches
Based on the cone shown, which statements are correct? (check 2 that apply)
V=1/3pi(15)^2h can be used to find the volume of the cone
if the volume is 47cm^3, then the height of the cone is approximately 5cm
Disregarding the 17 inch slant height, if the height of the cone is 20cm, then the volume of the cone is approximately 289cm^3
disregarding the 17 inch slant height, if the height of the cone is 10cm, then the volume of the cone is approximately 2356cm^3
h=3V/pi(17^2) can be used to find the height of the cone
Based on the cone shown, which statements are correct? (check 2 that apply)
V=1/3pi(15)^2h can be used to find the volume of the cone
if the volume is 47cm^3, then the height of the cone is approximately 5cm
Disregarding the 17 inch slant height, if the height of the cone is 20cm, then the volume of the cone is approximately 289cm^3
disregarding the 17 inch slant height, if the height of the cone is 10cm, then the volume of the cone is approximately 2356cm^3
h=3V/pi(17^2) can be used to find the height of the cone
Answers
GPT-4o mini
Let's analyze each statement one by one for correctness.
1. **V = (1/3)π(15)^2h can be used to find the volume of the cone**: This statement is **correct**. The formula for the volume of a cone is indeed V = (1/3)πr²h, where r is the radius. Here, the radius is given as 15 inches, so this formula applies.
2. **If the volume is 47 cm³, then the height of the cone is approximately 5 cm**: To check the correctness, we can use the formula for volume:
\[
V = \frac{1}{3}πr²h
\]
Substituting r = 15 cm and V = 47 cm³:
\[
47 = \frac{1}{3}π(15)^2h \\
47 = \frac{1}{3}π(225)h \\
141 = 225πh \\
h = \frac{141}{225π} \approx 0.2 \text{ cm}
\]
This statement is **incorrect**, the height would not be approximately 5 cm.
3. **Disregarding the 17 inch slant height, if the height of the cone is 20 cm, then the volume of the cone is approximately 289 cm³**: Let's calculate the volume using h = 20 cm:
\[
V = \frac{1}{3}π(15)^2(20) = \frac{1}{3}π(225)(20) = \frac{4500}{3}π = 1500π \approx 4712 \text{ cm}^3
\]
This statement is **incorrect**; 20 cm height would lead to a volume much larger than 289 cm³.
4. **Disregarding the 17 inch slant height, if the height of the cone is 10 cm, then the volume of the cone is approximately 2356 cm³**: Let's calculate for h = 10 cm:
\[
V = \frac{1}{3}π(15)^2(10) = \frac{1}{3}π(225)(10) = \frac{2250}{3}π = 750π \approx 2356 \text{ cm}^3
\]
This statement is **correct**.
5. **h = 3V/(π(17)^2) can be used to find the height of the cone**: This formula is incorrect for the cone's volume where h is based on the radius (15 inches) and not the slant height (17 inches). The correct formula for h should be derived directly using the radius (15 inches). Therefore, this statement is **incorrect**.
Based on the evaluation, the two correct statements are:
1. V = (1/3)π(15)²h can be used to find the volume of the cone.
2. Disregarding the 17 inch slant height, if the height of the cone is 10 cm, then the volume of the cone is approximately 2356 cm³.
1. **V = (1/3)π(15)^2h can be used to find the volume of the cone**: This statement is **correct**. The formula for the volume of a cone is indeed V = (1/3)πr²h, where r is the radius. Here, the radius is given as 15 inches, so this formula applies.
2. **If the volume is 47 cm³, then the height of the cone is approximately 5 cm**: To check the correctness, we can use the formula for volume:
\[
V = \frac{1}{3}πr²h
\]
Substituting r = 15 cm and V = 47 cm³:
\[
47 = \frac{1}{3}π(15)^2h \\
47 = \frac{1}{3}π(225)h \\
141 = 225πh \\
h = \frac{141}{225π} \approx 0.2 \text{ cm}
\]
This statement is **incorrect**, the height would not be approximately 5 cm.
3. **Disregarding the 17 inch slant height, if the height of the cone is 20 cm, then the volume of the cone is approximately 289 cm³**: Let's calculate the volume using h = 20 cm:
\[
V = \frac{1}{3}π(15)^2(20) = \frac{1}{3}π(225)(20) = \frac{4500}{3}π = 1500π \approx 4712 \text{ cm}^3
\]
This statement is **incorrect**; 20 cm height would lead to a volume much larger than 289 cm³.
4. **Disregarding the 17 inch slant height, if the height of the cone is 10 cm, then the volume of the cone is approximately 2356 cm³**: Let's calculate for h = 10 cm:
\[
V = \frac{1}{3}π(15)^2(10) = \frac{1}{3}π(225)(10) = \frac{2250}{3}π = 750π \approx 2356 \text{ cm}^3
\]
This statement is **correct**.
5. **h = 3V/(π(17)^2) can be used to find the height of the cone**: This formula is incorrect for the cone's volume where h is based on the radius (15 inches) and not the slant height (17 inches). The correct formula for h should be derived directly using the radius (15 inches). Therefore, this statement is **incorrect**.
Based on the evaluation, the two correct statements are:
1. V = (1/3)π(15)²h can be used to find the volume of the cone.
2. Disregarding the 17 inch slant height, if the height of the cone is 10 cm, then the volume of the cone is approximately 2356 cm³.