Questions: A conical oil cup with a radius of 5.8 cm must be designed to hold 65 cu cm of oil. What should be the altitude of the cup? Round to the nearest tenth if necessary. The altitude of the cup is approximately cm. (Type an integer or a decimal. Round to nearest tenth as needed.)

Solution Steps

To find the altitude of the conical oil cup, we can use the formula for the volume of a cone, which is \( V = \frac{1}{3} \pi r^2 h \), where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height (altitude) of the cone. We are given the volume and the radius, and we need to solve for the height \( h \).

1. Substitute the given values into the volume formula.
2. Solve the equation for \( h \).

The volume \( V \) of a cone is given by the formula: \[ V = \frac{1}{3} \pi r^2 h \] where \( r \) is the radius and \( h \) is the height (altitude) of the cone.

We know the volume \( V = 65 \, \text{cm}^3 \) and the radius \( r = 5.8 \, \text{cm} \). Substituting these values into the volume formula gives: \[ 65 = \frac{1}{3} \pi (5.8)^2 h \]

Rearranging the equation to solve for \( h \): \[ h = \frac{3 \times 65}{\pi (5.8)^2} \] Calculating the right-hand side yields: \[ h \approx 1.8451375685445655 \] Rounding this to the nearest tenth gives: \[ h \approx 1.8 \]

Final Answer