Question
can you make sense out of this
SA = 4πr^2
78.5 = 4(3.14)r^2
78.5 = 12.56r^2
6.25 = r^2
2.5 = r
SA = 4πr^2
78.5 = 4(3.14)r^2
78.5 = 12.56r^2
6.25 = r^2
2.5 = r
Answers
GPT-4o mini
Yes! Let's break it down step by step.
The formula you started with is for the surface area (SA) of a sphere:
\[ SA = 4\pi r^2 \]
You set the surface area equal to \( 78.5 \):
\[ 78.5 = 4(3.14)r^2 \]
Next, you substituted \( \pi \) with \( 3.14 \) and performed the multiplication:
\[ 78.5 = 12.56r^2 \]
After that, you divided both sides by \( 12.56 \) to solve for \( r^2 \):
\[ r^2 = \frac{78.5}{12.56} \]
Calculating this gives:
\[ r^2 = 6.25 \]
Next, you take the square root of both sides to find \( r \):
\[ r = \sqrt{6.25} \]
Which results in:
\[ r = 2.5 \]
So, the radius \( r \) of the sphere with a surface area of \( 78.5 \) (when using \( \pi \approx 3.14 \)) is \( 2.5 \) units. Everything you've calculated and shown makes sense!
The formula you started with is for the surface area (SA) of a sphere:
\[ SA = 4\pi r^2 \]
You set the surface area equal to \( 78.5 \):
\[ 78.5 = 4(3.14)r^2 \]
Next, you substituted \( \pi \) with \( 3.14 \) and performed the multiplication:
\[ 78.5 = 12.56r^2 \]
After that, you divided both sides by \( 12.56 \) to solve for \( r^2 \):
\[ r^2 = \frac{78.5}{12.56} \]
Calculating this gives:
\[ r^2 = 6.25 \]
Next, you take the square root of both sides to find \( r \):
\[ r = \sqrt{6.25} \]
Which results in:
\[ r = 2.5 \]
So, the radius \( r \) of the sphere with a surface area of \( 78.5 \) (when using \( \pi \approx 3.14 \)) is \( 2.5 \) units. Everything you've calculated and shown makes sense!
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? Round your answer to the nearest hundredth. Use 3.14 for pi.(1 point)
GPT-4o mini
To find the surface area of a cone, you need to calculate both the lateral surface area and the area of the base (if applicable). However, since it seems you're only asking for the area of the peak (which is the lateral surface area), we will only calculate that.
The formula for the lateral surface area \( A \) of a cone is:
\[
A = \pi r l
\]
where:
- \( r \) is the radius of the cone,
- \( l \) is the slant height of the cone.
Given:
- The diameter of the cone is 14 inches, so the radius \( r \) is:
\[
r = \frac{\text{diameter}}{2} = \frac{14}{2} = 7 \text{ inches}
\]
- The slant height \( l \) is given as 20 inches.
Now we can substitute these values into the formula:
\[
A = 3.14 \times 7 \times 20
\]
Calculating this step-by-step:
1. Calculate \( 7 \times 20 = 140 \).
2. Then multiply by \( 3.14 \):
\[
A = 3.14 \times 140 = 439.6
\]
Thus, the lateral surface area of the cone peak is approximately **439.60 square inches** when rounded to the nearest hundredth.
The formula for the lateral surface area \( A \) of a cone is:
\[
A = \pi r l
\]
where:
- \( r \) is the radius of the cone,
- \( l \) is the slant height of the cone.
Given:
- The diameter of the cone is 14 inches, so the radius \( r \) is:
\[
r = \frac{\text{diameter}}{2} = \frac{14}{2} = 7 \text{ inches}
\]
- The slant height \( l \) is given as 20 inches.
Now we can substitute these values into the formula:
\[
A = 3.14 \times 7 \times 20
\]
Calculating this step-by-step:
1. Calculate \( 7 \times 20 = 140 \).
2. Then multiply by \( 3.14 \):
\[
A = 3.14 \times 140 = 439.6
\]
Thus, the lateral surface area of the cone peak is approximately **439.60 square inches** when rounded to the nearest hundredth.