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Sagot :
Sure! Let's solve the problem step by step.
1. Understand the problem:
- We are given the volume of a miniature basketball, which is approximately 113 cubic inches.
- We need to find the radius of the sphere (the basketball), and round it to the nearest inch.
2. Recall the formula for the volume of a sphere:
- The volume [tex]\( V \)[/tex] of a sphere is given by the formula:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
- Here, [tex]\( V \)[/tex] is the volume and [tex]\( r \)[/tex] is the radius of the sphere.
3. Rearrange the formula to solve for the radius:
- We need to isolate [tex]\( r \)[/tex] in the formula. Start by multiplying both sides of the equation by [tex]\(\frac{3}{4 \pi}\)[/tex]:
[tex]\[ r^3 = \frac{3V}{4\pi} \][/tex]
- Then, take the cube root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \left( \frac{3V}{4\pi} \right)^{\frac{1}{3}} \][/tex]
4. Substitute the known volume into the formula:
- Here, [tex]\( V = 113 \)[/tex] cubic inches:
[tex]\[ r = \left( \frac{3 \times 113}{4\pi} \right)^{\frac{1}{3}} \][/tex]
5. Calculate the radius:
- Using the volume [tex]\( V = 113 \)[/tex] cubic inches:
[tex]\[ r \approx 2.9991391179501163 \text{ inches} \][/tex]
6. Round the radius to the nearest inch:
- The radius [tex]\( r \approx 2.9991391179501163 \)[/tex] inches rounds to [tex]\( 3 \)[/tex] inches.
Therefore, the length of the basketball's radius, rounded to the nearest inch, is [tex]\( 3 \)[/tex] inches.
If you're entering a student-produced response on the SAT for this problem, here's how you can do it:
- Feedback System: With the given numeric result [tex]\( 3 \)[/tex], you will directly enter this value into the appropriate section of the answer sheet where numerical responses are required.
- Ensure that there are no extraneous marks or spaces around the number you entered.
That's how you can find and report the radius of the basketball.
1. Understand the problem:
- We are given the volume of a miniature basketball, which is approximately 113 cubic inches.
- We need to find the radius of the sphere (the basketball), and round it to the nearest inch.
2. Recall the formula for the volume of a sphere:
- The volume [tex]\( V \)[/tex] of a sphere is given by the formula:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
- Here, [tex]\( V \)[/tex] is the volume and [tex]\( r \)[/tex] is the radius of the sphere.
3. Rearrange the formula to solve for the radius:
- We need to isolate [tex]\( r \)[/tex] in the formula. Start by multiplying both sides of the equation by [tex]\(\frac{3}{4 \pi}\)[/tex]:
[tex]\[ r^3 = \frac{3V}{4\pi} \][/tex]
- Then, take the cube root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \left( \frac{3V}{4\pi} \right)^{\frac{1}{3}} \][/tex]
4. Substitute the known volume into the formula:
- Here, [tex]\( V = 113 \)[/tex] cubic inches:
[tex]\[ r = \left( \frac{3 \times 113}{4\pi} \right)^{\frac{1}{3}} \][/tex]
5. Calculate the radius:
- Using the volume [tex]\( V = 113 \)[/tex] cubic inches:
[tex]\[ r \approx 2.9991391179501163 \text{ inches} \][/tex]
6. Round the radius to the nearest inch:
- The radius [tex]\( r \approx 2.9991391179501163 \)[/tex] inches rounds to [tex]\( 3 \)[/tex] inches.
Therefore, the length of the basketball's radius, rounded to the nearest inch, is [tex]\( 3 \)[/tex] inches.
If you're entering a student-produced response on the SAT for this problem, here's how you can do it:
- Feedback System: With the given numeric result [tex]\( 3 \)[/tex], you will directly enter this value into the appropriate section of the answer sheet where numerical responses are required.
- Ensure that there are no extraneous marks or spaces around the number you entered.
That's how you can find and report the radius of the basketball.
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