Question
the diameter is 25
the length is 22
What is the surface area of the cone? Use 3.14 for pi.
(1 point)
Responses
3,689.5 square centimeters
3,689.5 square centimeters
2,383.26 square centimeters
2,383.26 square centimeters
863.5 square centimeters
863.5 square centimeters
1,354.125 square centimeters
the length is 22
What is the surface area of the cone? Use 3.14 for pi.
(1 point)
Responses
3,689.5 square centimeters
3,689.5 square centimeters
2,383.26 square centimeters
2,383.26 square centimeters
863.5 square centimeters
863.5 square centimeters
1,354.125 square centimeters
Answers
GPT-4o mini
To find the surface area of a cone, you can use the formula:
\[
\text{Surface Area} = \pi r (r + l)
\]
where:
- \( r \) is the radius of the base of the cone,
- \( l \) is the slant height of the cone, and
- \( \pi \) is approximately \( 3.14 \).
Given that the diameter is 25 cm, the radius \( r \) is half of that:
\[
r = \frac{25}{2} = 12.5 \text{ cm}
\]
The length given is the height \( h \) of the cone, which is 22 cm. We need to find the slant height \( l \) using the Pythagorean theorem:
\[
l = \sqrt{r^2 + h^2}
\]
Now, calculating \( l \):
\[
l = \sqrt{(12.5)^2 + (22)^2} = \sqrt{156.25 + 484} = \sqrt{640.25} \approx 25.3 \text{ cm}
\]
Now we can substitute \( r \) and \( l \) back into the surface area formula:
\[
\text{Surface Area} = 3.14 \times 12.5 \times (12.5 + 25.3)
\]
Calculating \( 12.5 + 25.3 \):
\[
12.5 + 25.3 = 37.8
\]
Now substitute back into the surface area equation:
\[
\text{Surface Area} \approx 3.14 \times 12.5 \times 37.8
\]
Calculating this step-by-step:
1. Calculate \( 12.5 \times 37.8 \):
\[
12.5 \times 37.8 = 472.5
\]
2. Then calculate \( 3.14 \times 472.5 \):
\[
3.14 \times 472.5 \approx 1,484.85 \text{ square centimeters}
\]
It appears my calculations did not yield an option. Let’s double-check the calculation step.
1. First, calculate the apex slant height again using confirmed values:
\[
l \approx 25.3 \text{ cm}
\]
2. now recalculate:
\[
\text{Surface Area} = \pi r (r + l) = 3.14 \times 12.5 \times (12.5 + 25.3)
\]
Now computing the result:
- \( (12.5 + 25.3 = 37.8) \)
- Then compute \( 3.14 \times 12.5 = 39.25 \)
- Finally compute \( 39.25 \times 37.8 \approx 1485.45 \text{ square centimeters} \).
Based on the multiple-choice responses, it may be that there has been either confusion or change in expected parameters, but importantly, the rounded calculation doesn't align with provided options. I would validate the steps or the options given initially.
None of the provided answers (3,689.5, 2,383.26, 863.5, or 1,354.125) matches this calculation accurately. Therefore, you may need to check the values or consult other resources for confirmation on your response requirements.
\[
\text{Surface Area} = \pi r (r + l)
\]
where:
- \( r \) is the radius of the base of the cone,
- \( l \) is the slant height of the cone, and
- \( \pi \) is approximately \( 3.14 \).
Given that the diameter is 25 cm, the radius \( r \) is half of that:
\[
r = \frac{25}{2} = 12.5 \text{ cm}
\]
The length given is the height \( h \) of the cone, which is 22 cm. We need to find the slant height \( l \) using the Pythagorean theorem:
\[
l = \sqrt{r^2 + h^2}
\]
Now, calculating \( l \):
\[
l = \sqrt{(12.5)^2 + (22)^2} = \sqrt{156.25 + 484} = \sqrt{640.25} \approx 25.3 \text{ cm}
\]
Now we can substitute \( r \) and \( l \) back into the surface area formula:
\[
\text{Surface Area} = 3.14 \times 12.5 \times (12.5 + 25.3)
\]
Calculating \( 12.5 + 25.3 \):
\[
12.5 + 25.3 = 37.8
\]
Now substitute back into the surface area equation:
\[
\text{Surface Area} \approx 3.14 \times 12.5 \times 37.8
\]
Calculating this step-by-step:
1. Calculate \( 12.5 \times 37.8 \):
\[
12.5 \times 37.8 = 472.5
\]
2. Then calculate \( 3.14 \times 472.5 \):
\[
3.14 \times 472.5 \approx 1,484.85 \text{ square centimeters}
\]
It appears my calculations did not yield an option. Let’s double-check the calculation step.
1. First, calculate the apex slant height again using confirmed values:
\[
l \approx 25.3 \text{ cm}
\]
2. now recalculate:
\[
\text{Surface Area} = \pi r (r + l) = 3.14 \times 12.5 \times (12.5 + 25.3)
\]
Now computing the result:
- \( (12.5 + 25.3 = 37.8) \)
- Then compute \( 3.14 \times 12.5 = 39.25 \)
- Finally compute \( 39.25 \times 37.8 \approx 1485.45 \text{ square centimeters} \).
Based on the multiple-choice responses, it may be that there has been either confusion or change in expected parameters, but importantly, the rounded calculation doesn't align with provided options. I would validate the steps or the options given initially.
None of the provided answers (3,689.5, 2,383.26, 863.5, or 1,354.125) matches this calculation accurately. Therefore, you may need to check the values or consult other resources for confirmation on your response requirements.