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A cylinder and cone have equal volumes and radii of equal length. If the height of the cone is 24 centimeters, then what is the height of the cylinder in centimeters?

A cylinder and cone have equal volumes and radii of equal length. If the height of the cone is 2424 centimeters, then what is the height of the cylinder in centimeters?

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Pythagorean Theorem and its converseright chevron icon

Full solution

Q. A cylinder and cone have equal volumes and radii of equal length. If the height of the cone is 2424 centimeters, then what is the height of the cylinder in centimeters?
  1. Volume of a Cone Formula: The volume of a cone is given by the formula V=13πr2hV = \frac{1}{3}\pi r^2 h, where rr is the radius and hh is the height of the cone. Since the cylinder and the cone have equal volumes, we can set the volume of the cone equal to the volume of the cylinder, which is V=πr2HV = \pi r^2 H, where HH is the height of the cylinder.
  2. Setting the Volumes Equal: Let's denote the common radius of the cylinder and the cone as rr, the height of the cone as hc=24 cmh_c = 24\,\text{cm}, and the height of the cylinder as hcylh_{\text{cyl}}. We can write the volume of the cone as Vcone=13πr2hcV_{\text{cone}} = \frac{1}{3}\pi r^2 h_c.
  3. Denoting the Common Measurements: The volume of the cylinder is Vcylinder=πr2hcylV_{\text{cylinder}} = \pi r^2 h_{\text{cyl}}. Since the volumes are equal, we have Vcone=VcylinderV_{\text{cone}} = V_{\text{cylinder}}.
  4. Writing the Volume Equations: Substituting the volume formulas, we get (13)πr2hc=πr2hcyl(\frac{1}{3})\pi r^2 h_c = \pi r^2 h_{\text{cyl}}. We can cancel πr2\pi r^2 from both sides since they are not zero.
  5. Canceling Out Terms: After canceling, we are left with (13)hc=hcyl(\frac{1}{3})h_c = h_{\text{cyl}}. We can now substitute the known value of hch_c, which is 2424 cm.
  6. Substituting the Known Value: So, (13)×24 cm=hcyl(\frac{1}{3}) \times 24 \, \text{cm} = h_{\text{cyl}}. This simplifies to 8 cm=hcyl8 \, \text{cm} = h_{\text{cyl}}.

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