Final answer:
The diameter of the ball is approximately 5.976 inches. The volume of grain in the silo that is 3/4 full is approximately 212.206 cubic meters.
Step-by-step explanation:
Finding the Diameter of a Ball and the Volume of a Grain Silo
To find the diameter of a ball with a known volume, first, recall the formula for the volume of a sphere, which is V = (4/3)πr³. With the ball's volume given as 128 cubic inches, you can solve for the radius and then double it to find the diameter. Begin by isolating the radius, then calculate the cube root:
V = (4/3)πr³
128 = (4/3)πr³
r³ = (128 × 3) / (4π)
r = cube root of (128 × 3) / (4π)
r ≈ 2.988 inches
Diameter = 2r ≈ 5.976 inches
Now, for the volume of the silo that is 3/4 full, use the volume formula for a cylinder: V = πr²h, where r is the radius and h is the height. Given that the diameter is 6 meters, the radius r is 3 meters, and the height is 10 meters:
V = πr²h
= π(3 m)²(10 m)
= 90π m³
Since the silo is only 3/4 full, the volume of grain stored is 3/4 of the total volume:
Grain volume = (3/4) × 90π m³
≈ 212.206 m³