Question
Eli+is+making+a+model+castle+out+of+clay.+One+of+the+roof+peaks+is+in+the+shape+of+a+cone+with+a+diameter+of+14+inches+and+a+slant+height+of+20+inches.+What+is+the+surface+area+of+the+cone+peak%3F+Round+your+answer+to+the+nearest+hundredth.+Use+3.14+for+pi
Answers
Answered by
GPT-4o mini
To find the surface area of the cone peak, we need to calculate the lateral surface area and the base area of the cone. The formula for the lateral surface area \( A_L \) of a cone is given by:
\[
A_L = \pi r l
\]
where:
- \( r \) is the radius of the base of the cone,
- \( l \) is the slant height of the cone.
The base area \( A_B \) of the cone is given by the formula:
\[
A_B = \pi r^2
\]
The total surface area \( A_{total} \) of the cone is the sum of the lateral surface area and the base area:
\[
A_{total} = A_L + A_B
\]
Given the information:
- The diameter of the cone is 14 inches, so the radius \( r \) is:
\[
r = \frac{14}{2} = 7 \text{ inches}
\]
- The slant height \( l \) is 20 inches.
### Step 1: Calculate the lateral surface area
Using the formula for lateral surface area:
\[
A_L = \pi r l = 3.14 \cdot 7 \cdot 20
\]
Calculating \( 7 \cdot 20 = 140 \):
\[
A_L = 3.14 \cdot 140 = 439.6 \text{ square inches}
\]
### Step 2: Calculate the base area
Using the formula for the base area:
\[
A_B = \pi r^2 = 3.14 \cdot 7^2 = 3.14 \cdot 49
\]
Calculating \( 3.14 \cdot 49 = 153.86 \text{ square inches} \):
### Step 3: Calculate the total surface area
Now, we can find the total surface area:
\[
A_{total} = A_L + A_B = 439.6 + 153.86
\]
Adding these together:
\[
A_{total} = 593.46 \text{ square inches}
\]
### Conclusion
Rounding to the nearest hundredth, the surface area of the cone peak is:
\[
\boxed{593.46} \text{ square inches}
\]
\[
A_L = \pi r l
\]
where:
- \( r \) is the radius of the base of the cone,
- \( l \) is the slant height of the cone.
The base area \( A_B \) of the cone is given by the formula:
\[
A_B = \pi r^2
\]
The total surface area \( A_{total} \) of the cone is the sum of the lateral surface area and the base area:
\[
A_{total} = A_L + A_B
\]
Given the information:
- The diameter of the cone is 14 inches, so the radius \( r \) is:
\[
r = \frac{14}{2} = 7 \text{ inches}
\]
- The slant height \( l \) is 20 inches.
### Step 1: Calculate the lateral surface area
Using the formula for lateral surface area:
\[
A_L = \pi r l = 3.14 \cdot 7 \cdot 20
\]
Calculating \( 7 \cdot 20 = 140 \):
\[
A_L = 3.14 \cdot 140 = 439.6 \text{ square inches}
\]
### Step 2: Calculate the base area
Using the formula for the base area:
\[
A_B = \pi r^2 = 3.14 \cdot 7^2 = 3.14 \cdot 49
\]
Calculating \( 3.14 \cdot 49 = 153.86 \text{ square inches} \):
### Step 3: Calculate the total surface area
Now, we can find the total surface area:
\[
A_{total} = A_L + A_B = 439.6 + 153.86
\]
Adding these together:
\[
A_{total} = 593.46 \text{ square inches}
\]
### Conclusion
Rounding to the nearest hundredth, the surface area of the cone peak is:
\[
\boxed{593.46} \text{ square inches}
\]