At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Get immediate and reliable solutions to your questions from a knowledgeable community of professionals on our platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To determine which expression represents the volume of a sphere with a radius of 6 units, we need to use the formula for the volume of a sphere:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
Given \( r = 6 \), substituting this into the formula gives:
[tex]\[ V = \frac{4}{3} \pi (6)^3 \][/tex]
Let's assess each of the given expressions one by one to see if they match this formula.
1. Expression: \( \frac{3}{4} \pi (6)^2 \)
- This expression calculates the area of a circle (not the volume of a sphere), but adjusted by the fraction.
- Volume check: \(\frac{3}{4} \pi (6)^2 = \frac{3}{4} \pi (36) = 27 \pi\)
2. Expression: \( \frac{4}{3} \pi (6)^3 \)
- This is the exact formula for the volume of a sphere with radius 6.
- Volume calculation: \( \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288 \pi\)
3. Expression: \( \frac{3}{4} \pi (12)^2 \)
- This calculates the area of a circle with diameter 12, adjusted by the fraction.
- Volume check: \(\frac{3}{4} \pi (12)^2 = \frac{3}{4} \pi (144) = 108 \pi\)
4. Expression: \( \frac{4}{3} \pi (12)^3 \)
- This calculates the volume of a sphere with radius 12, not 6.
- Volume check: \(\frac{4}{3} \pi (12)^3 = \frac{4}{3} \pi (1728) = 2304 \pi\)
From our evaluations, only the expression \(\frac{4}{3} \pi (6)^3\) matches both the necessary formula and the correct radius value.
Thus, the correct expression representing the volume of the sphere is:
[tex]\[ \frac{4}{3} \pi (6)^3 \][/tex]
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
Given \( r = 6 \), substituting this into the formula gives:
[tex]\[ V = \frac{4}{3} \pi (6)^3 \][/tex]
Let's assess each of the given expressions one by one to see if they match this formula.
1. Expression: \( \frac{3}{4} \pi (6)^2 \)
- This expression calculates the area of a circle (not the volume of a sphere), but adjusted by the fraction.
- Volume check: \(\frac{3}{4} \pi (6)^2 = \frac{3}{4} \pi (36) = 27 \pi\)
2. Expression: \( \frac{4}{3} \pi (6)^3 \)
- This is the exact formula for the volume of a sphere with radius 6.
- Volume calculation: \( \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288 \pi\)
3. Expression: \( \frac{3}{4} \pi (12)^2 \)
- This calculates the area of a circle with diameter 12, adjusted by the fraction.
- Volume check: \(\frac{3}{4} \pi (12)^2 = \frac{3}{4} \pi (144) = 108 \pi\)
4. Expression: \( \frac{4}{3} \pi (12)^3 \)
- This calculates the volume of a sphere with radius 12, not 6.
- Volume check: \(\frac{4}{3} \pi (12)^3 = \frac{4}{3} \pi (1728) = 2304 \pi\)
From our evaluations, only the expression \(\frac{4}{3} \pi (6)^3\) matches both the necessary formula and the correct radius value.
Thus, the correct expression representing the volume of the sphere is:
[tex]\[ \frac{4}{3} \pi (6)^3 \][/tex]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.