Get the answers you need from a community of experts on IDNLearn.com. Our experts provide accurate and detailed responses to help you navigate any topic or issue with confidence.
Sagot :
To solve for [tex]\(\frac{V}{\pi}\)[/tex] given the surface area of the sphere, we start by using the formula for the surface area of a sphere:
[tex]\[ A = 4 \pi r^2 \][/tex]
Here, the surface area [tex]\( A \)[/tex] is given as [tex]\( 2500 \pi \)[/tex] square inches. Substituting [tex]\( A = 2500 \pi \)[/tex] into the formula, we have:
[tex]\[ 2500 \pi = 4 \pi r^2 \][/tex]
Next, we need to solve for the radius [tex]\( r \)[/tex]. First, divide both sides by [tex]\( 4 \pi \)[/tex]:
[tex]\[ r^2 = \frac{2500 \pi}{4 \pi} \][/tex]
Simplify the right-hand side:
[tex]\[ r^2 = \frac{2500}{4} = 625 \][/tex]
Now solve for [tex]\( r \)[/tex] by taking the square root of both sides:
[tex]\[ r = \sqrt{625} \][/tex]
[tex]\[ r = 25 \][/tex]
With the radius [tex]\( r \)[/tex] now known, we can find the volume [tex]\( V \)[/tex] of the sphere using the formula:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
Substitute [tex]\( r = 25 \)[/tex] into this formula:
[tex]\[ V = \frac{4}{3} \pi (25)^3 \][/tex]
Now calculate [tex]\( (25)^3 \)[/tex]:
[tex]\[ 25^3 = 25 \times 25 \times 25 \][/tex]
[tex]\[ 25 \times 25 = 625 \][/tex]
[tex]\[ 625 \times 25 = 15625 \][/tex]
So:
[tex]\[ V = \frac{4}{3} \pi \times 15625 \][/tex]
Now simplify the expression inside the volume formula:
[tex]\[ V = \frac{4 \times 15625}{3} \pi \][/tex]
[tex]\[ V = \frac{62500}{3} \pi \][/tex]
Next, we need [tex]\(\frac{V}{\pi}\)[/tex]:
[tex]\[ \frac{V}{\pi} = \frac{\frac{62500}{3} \pi}{\pi} \][/tex]
The [tex]\(\pi\)[/tex] terms cancel out:
[tex]\[ \frac{V}{\pi} = \frac{62500}{3} \][/tex]
Now compute [tex]\(\frac{62500}{3}\)[/tex]:
[tex]\[ \frac{62500}{3} \approx 20833.33 \][/tex]
Therefore, the value of [tex]\(\frac{V}{\pi}\)[/tex] is approximately:
[tex]\[ 20833.33 \][/tex]
So, the correct answer is:
[tex]\[ 20833.33 \ln^3 \][/tex]
[tex]\[ A = 4 \pi r^2 \][/tex]
Here, the surface area [tex]\( A \)[/tex] is given as [tex]\( 2500 \pi \)[/tex] square inches. Substituting [tex]\( A = 2500 \pi \)[/tex] into the formula, we have:
[tex]\[ 2500 \pi = 4 \pi r^2 \][/tex]
Next, we need to solve for the radius [tex]\( r \)[/tex]. First, divide both sides by [tex]\( 4 \pi \)[/tex]:
[tex]\[ r^2 = \frac{2500 \pi}{4 \pi} \][/tex]
Simplify the right-hand side:
[tex]\[ r^2 = \frac{2500}{4} = 625 \][/tex]
Now solve for [tex]\( r \)[/tex] by taking the square root of both sides:
[tex]\[ r = \sqrt{625} \][/tex]
[tex]\[ r = 25 \][/tex]
With the radius [tex]\( r \)[/tex] now known, we can find the volume [tex]\( V \)[/tex] of the sphere using the formula:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
Substitute [tex]\( r = 25 \)[/tex] into this formula:
[tex]\[ V = \frac{4}{3} \pi (25)^3 \][/tex]
Now calculate [tex]\( (25)^3 \)[/tex]:
[tex]\[ 25^3 = 25 \times 25 \times 25 \][/tex]
[tex]\[ 25 \times 25 = 625 \][/tex]
[tex]\[ 625 \times 25 = 15625 \][/tex]
So:
[tex]\[ V = \frac{4}{3} \pi \times 15625 \][/tex]
Now simplify the expression inside the volume formula:
[tex]\[ V = \frac{4 \times 15625}{3} \pi \][/tex]
[tex]\[ V = \frac{62500}{3} \pi \][/tex]
Next, we need [tex]\(\frac{V}{\pi}\)[/tex]:
[tex]\[ \frac{V}{\pi} = \frac{\frac{62500}{3} \pi}{\pi} \][/tex]
The [tex]\(\pi\)[/tex] terms cancel out:
[tex]\[ \frac{V}{\pi} = \frac{62500}{3} \][/tex]
Now compute [tex]\(\frac{62500}{3}\)[/tex]:
[tex]\[ \frac{62500}{3} \approx 20833.33 \][/tex]
Therefore, the value of [tex]\(\frac{V}{\pi}\)[/tex] is approximately:
[tex]\[ 20833.33 \][/tex]
So, the correct answer is:
[tex]\[ 20833.33 \ln^3 \][/tex]
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.