The bottom portion of a loading bin is cone shaped. The base radius of this part of the bin is 3.5 feet and the slant height is 6.5 feet. What is the capacity and lateral surface area of this part of the bin? Round your answer to the nearest hundredth.
To find the capacity of the cone-shaped bottom portion of the loading bin, you can use the formula for the volume of a cone:
Volume = (1/3) * π * r^2 * h
Where r is the base radius and h is the height. In this case, the height of the cone is equal to the slant height.
Volume = (1/3) * π * (3.5^2) * 6.5
Calculating this using a calculator, we get the volume to be approximately 100.55 cubic feet. Therefore, the capacity of the cone-shaped bottom portion of the loading bin is approximately 100.55 cubic feet.
To find the lateral surface area of the cone-shaped bottom portion of the loading bin, you can use the formula for the lateral surface area of a cone:
Lateral Surface Area = π * r * l
Where r is the base radius and l is the slant height. In this case, we already have the values for r and l.
Lateral Surface Area = π * 3.5 * 6.5
Calculating this using a calculator, we get the lateral surface area to be approximately 71.98 square feet, rounded to the nearest hundredth.
Therefore, the capacity of the cone-shaped bottom portion of the loading bin is approximately 100.55 cubic feet, and the lateral surface area is approximately 71.98 square feet.