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The volume of the cone is 25π.

Match the Question with the correct answer.

(5 points)
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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?
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 sphere if the radius is the same value as the cone.
The volume of the cylinder

1 answer

Let's analyze each question and provide the correct answers step by step.

  1. 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 formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \). The formula for the volume of a cone is \( V = \frac{1}{3} \pi r^2 h \). If we consider the cone with a radius r and height h, we can find the ratio of their volumes.

    Assuming they have the same height and radius: Volume of the cone \( V_c = \frac{1}{3} \pi r^2 h \). We already know \( V_c = 25\pi \).

    If \( h = 3r \), we can express the volume of the cone as:

    \[ 25\pi = \frac{1}{3} \pi r^2 (3r) \quad \Rightarrow \quad 25 = r^3 \quad \Rightarrow \quad r = 5 \]

    Now calculating the volume of the sphere: \[ V_s = \frac{4}{3} \pi (5)^3 = \frac{4}{3} \pi (125) = \frac{500}{3} \pi \]

    To find out how many times larger the volume of the sphere is than the cone:

    \[ \text{Times larger} = \frac{V_s}{V_c} = \frac{\frac{500}{3} \pi}{25 \pi} = \frac{500}{3 \times 25} = \frac{500}{75} = \frac{20}{3} \approx 6.67 \]

  2. 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?

    The volume of a cylinder is \( V = \pi r^2 h \). If we assume the cylinder has the same radius r and height h as the cone (with \( h = 3r \)):

    \[ V_{\text{cylinder}} = \pi (5^2)(15) = \pi (25)(15) = 375\pi \]

    Now calculate the ratio: \[ \text{Times larger} = \frac{V_{\text{cylinder}}}{V_c} = \frac{375 \pi}{25 \pi} = 15 \]

  3. 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.

    You would need to determine the required height to satisfy the equation. Knowing that \( V_c = 25\pi \), \( V_s = \frac{500}{3}\pi \), and \( V_{\text{cylinder}} \) needs to equal the combined sea volumes, we have:

    \[ 25\pi + \frac{500}{3}\pi = \frac{375}{3}\pi \]

    Thus we need to satisfy the equation \( V_{\text{cylinder}} = \frac{375}{3}\pi\).

    The height needs to be adjusted so each volume satisfies \( \pi r^2 h = ? \)

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

    Since we already calculated that: \[ V_s = \frac{500}{3} \pi \]

  5. The volume of the cylinder.

    The volume of the cylinder as calculated in question 2 is: \[ V_{\text{cylinder}} = 375\pi \]

Now matching:

  1. The volume of the sphere if the radius is the same value as the cone: \(\frac{500}{3} \pi\)
  2. The volume of the cylinder: \(375\pi\)
  3. How many times larger is the volume of the sphere than the volume of the cone: \(\frac{20}{3}\)
  4. How many times larger is the volume of the cylinder than the volume of the cone: \(15\)
  5. In order for the volume of the cone + the volume of the sphere = the volume of the cylinder, the height would have to be adjusted to not 3r.

Thus, fill these prompts with the calculated values for each respective input field.