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The volume of the cone is 25π.

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How many times larger is the volume of the sphere than the volume of the cone IF the cone and the sphere had the same dimensions?

The volume of the sphere if the radius is the same value as the cone.

How many times larger is the volume of the cylinder than the volume of the cone IF the cone and the cylinder had the same dimensions?

In order for the volume of the cone + the volume of the sphere = the volume of the cylinder, the height would have to be _______, not 3r for the cone and cylinder.

The volume of the cylinder

responces:
50 over 3 pi
2r
3r
75 over 2 pi
2
3
75 pi
25 pi
1/3
4r
100 over 3 pi
125 pi

1 answer

Let's break down the information provided:

  1. The volume of a cone is given as \( 25\pi \).
  2. The formulas for the volumes of the three shapes (cone, sphere, and cylinder) we are dealing with are:
    • Volume of a cone: \( V = \frac{1}{3} \pi r^2 h \)
    • Volume of a sphere: \( V = \frac{4}{3} \pi r^3 \)
    • Volume of a cylinder: \( V = \pi r^2 h \)

Step 1: Calculate the volume of the sphere with the same radius as the cone.

Given that the cone has the volume \( 25\pi \):

  • The formula for the volume of the cone as per its dimensions can be rearranged to find \( r \) if we assume the height \( h = 3r \) (as a potential starting point): \[ \frac{1}{3} \pi r^2 (3r) = 25\pi \] Simplifying gives: \[ \pi r^3 = 25\pi \implies r^3 = 25 \implies r = \sqrt[3]{25} \]

Now, substituting \( r \) into the volume of the sphere: \[ V_{sphere} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (25) = \frac{100}{3} \pi \]

Step 2: How many times larger is the volume of the sphere than the volume of the cone?

\[ \text{Ratio} = \frac{V_{sphere}}{V_{cone}} = \frac{\frac{100}{3}\pi}{25\pi} = \frac{100/3}{25} = \frac{100}{75} = \frac{4}{3} \] The sphere is \( \frac{4}{3} \) times larger than the cone.

Step 3: Calculate the volume of the cylinder with the same radius \( r \) and height \( h = 3r \).

Using the same height \( h \): \[ V_{cylinder} = \pi r^2 (3r) = 3\pi r^3 \] \[ = 3\pi (25) = 75\pi \]

Step 4: How many times larger is the volume of the cylinder than the volume of the cone?

\[ \text{Ratio} = \frac{V_{cylinder}}{V_{cone}} = \frac{75\pi}{25\pi} = 3 \] The cylinder is 3 times larger than the cone.

Step 5: Finding the height for the cone + sphere to equal the volume of the cylinder.

The volume equation will lead us to finding the height \( h \): \[ \text{Volume of cone} + \text{Volume of sphere} = \text{Volume of cylinder} \] \[ 25\pi + \frac{100}{3} \pi = \pi r^2 h \] Weighting these against the total shows: \[ 25 + \frac{100}{3} = \text{Expressed as a single fraction:} \] \[ = \frac{75}{3} + \frac{100}{3} = \frac{175}{3} \stackrel{=}{\text{for cylinder}} 75 = V_cylinder \]

Therefore, for the height consistent across the previous calculations, we can see:

  • Given approach required \( 5r \) (to account for shape overlap).

Answers matched with input:

  1. The volume of the sphere if the radius is the same value as the cone: 100 over 3 pi
  2. How many times larger is the volume of the sphere than the volume of the cone: 4/3
  3. How many times larger is the volume of the cylinder than the volume of the cone: 3
  4. In order for the volume of the cone + the volume of the sphere = the volume of the cylinder, the height would have to be not 3r for the cone and cylinder: 5r.
  5. The volume of the cylinder: 75 pi

Please map these correctly to the provided answers.