A sphere has five times the volume of a right cone that has a radius of 4.6 cm and a height of 7.7 cm. Determine the radius of the sphere to the tenth of a centimetre.
This question is multiple choice and these were the options:
A. 2.2cm
B. 3.4cm
C. 5.9cm
D. 9.3cm
I was sure it would be 5.9 cm, but I got that marked as incorrect.
This is how I got that answer.
I found the volume of the right cone and got 170.5 cm. Then I times that by 5 and got the volume of the sphere which is 852.5 cm. The formula for the sphere is 4/3*3.14*r^3. So I divided 4 by 3 and times that by 3.14. And I got 4.186666667. Then I divided 852.5/4.186666667 and I got 203.6225114. Then I triple rooted that and got 5.88 which I rounded off to 5.9. What am I doing wrong?
vol of cone = (1/3)π(4.6^2)(7.7) = 170.62 (you had that)
so volume of sphere = 853.1099
(4/3)πr^3 = 853.1099
r^3 = 3(853.1099)/(4π) = 203.665
r = cuberoot(50.91625) = 5.88 or their answer of 5.9
check:
sphere: V = (4/3)π(5.88^3) = 853.1099
cone = 170.62
and 853.1099/170.62 = 5.00005 , not bad!
You are correct, They are wrong!
Well, it seems like you did all the calculations correctly. I must say, your mathematical skills are as sharp as a pencil. However, it seems that the answer choices are not in agreement with your calculations. I guess the makers of the multiple-choice test must have misunderstood the concept of rounding. That can be quite obtuse sometimes! But don't worry, you did everything right, and the answer is indeed 5.9 cm. Sometimes, even the best of us get trapped in the web of multiple-choice confusion. Keep up the great work and don't let those tricky test makers get you down!
Your calculations are correct, and you found the correct volume for the sphere of approximately 852.5 cm³. However, the mistake was in rounding off the final answer to 5.9 cm.
To determine the radius of the sphere, we need to solve the equation for volume:
V_sphere = (4/3) * π * r^3
Given that V_sphere = 852.5 cm³, we can rearrange the equation:
852.5 = (4/3) * π * r^3
To find the radius, we can solve for 'r' by isolating it:
r^3 = (3 * 852.5) / (4 * π)
r^3 = 641.875 / π
r ≈ (641.875 / π)^(1/3)
Calculating this value gives us:
r ≈ (641.875 / 3.14)^(1/3)
r ≈ 9.256
Therefore, the correct answer is approximately 9.3 cm (option D).
Your calculations are correct, and your approach is sound. It seems like you made a rounding error when you rounded off the value of 5.88 to 5.9. However, the options given in the multiple-choice question are rounded to the nearest tenth of a centimeter. As a result, when you rounded the sphere's radius to 5.9 cm, it may have been too imprecise for the answer choices provided.
Solution:
To find the radius of the sphere, you can follow these steps:
1. Calculate the volume of the right cone using the formula V_cone = (1/3) * π * r^2 * h, where r is the cone's radius and h is its height.
V_cone = (1/3) * π * (4.6 cm)^2 * 7.7 cm ≈ 170.524 cm^3
2. Multiply the volume of the cone by 5 to determine the volume of the sphere: V_sphere = 5 * V_cone.
V_sphere = 5 * 170.524 cm^3 = 852.62 cm^3 (rounded to the nearest hundredth)
3. Calculate the radius of the sphere using the formula V_sphere = (4/3) * π * r^3.
852.62 cm^3 = (4/3) * π * r^3
4. Rearrange the equation to solve for r: r = (3 * V_sphere / (4 * π))^(1/3).
r = (3 * 852.62 cm^3 / (4 * π))^(1/3) ≈ 5.93 cm (rounded to the nearest hundredth)
Based on the given options, the closest answer to 5.9 cm would be 5.9 cm itself. Therefore, it appears that there might be a mistake in the provided answer choices.