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A cylinder and sphere have equal volumes and radii of equal length. If the height of the cylinder is 8 centimeters, then what is the length of the radius of each shape in centimeters?

A cylinder and sphere have equal volumes and radii of equal length. If the height of the cylinder is 88 centimeters, then what is the length of the radius of each shape in centimeters?

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Full solution

Q. A cylinder and sphere have equal volumes and radii of equal length. If the height of the cylinder is 88 centimeters, then what is the length of the radius of each shape in centimeters?
  1. Write formulas for volume: Write down the formulas for the volume of a cylinder and a sphere.\newlineThe volume of a cylinder is given by Vcylinder=πr2hV_{\text{cylinder}} = \pi r^2 h, where rr is the radius and hh is the height.\newlineThe volume of a sphere is given by Vsphere=43πr3V_{\text{sphere}} = \frac{4}{3}\pi r^3, where rr is the radius.
  2. Set volumes equal: Set the volumes of the cylinder and sphere equal to each other because they are given to have equal volumes. πr2h=43πr3\pi r^2 h = \frac{4}{3}\pi r^3
  3. Plug in cylinder height: Plug in the given height of the cylinder into the equation. πr2(8 cm)=(43)πr3\pi r^2(8 \text{ cm}) = \left(\frac{4}{3}\right)\pi r^3
  4. Simplify equation: Simplify the equation by dividing both sides by π\pi to eliminate the π\pi term.\newliner2(8 cm)=(43)r3r^2(8\,\text{cm}) = \left(\frac{4}{3}\right)r^3
  5. Divide by r2r^2: Divide both sides of the equation by r2r^2 to solve for rr.\newline8 cm=(43)r8\,\text{cm} = \left(\frac{4}{3}\right)r
  6. Isolate rr: Multiply both sides of the equation by 34\frac{3}{4} to isolate rr on one side.\newline(34)(8 cm)=r\left(\frac{3}{4}\right)(8\,\text{cm}) = r
  7. Calculate r: Calculate the value of r.\newliner = (34)(8 cm)=6 cm(\frac{3}{4})(8\,\text{cm}) = 6\,\text{cm}

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