Applying volume of solids | Solid geometry | High school geometry | Khan Academy

TL;DR

A cone-shaped grain hopper with specific dimensions is filled with grain, and various calculations are made regarding its volume, the number of boxes it can fill, and the time it takes to fill the boxes.

Transcript

• [Instructor] We're told that a cone-shaped grain hopper, and they highlight hopper in blue here, in case you want to know its definition on the exercise. It's something that would store grain, and then it can kind of fall out the bottom, has a radius of 10 meters at the top and is 8 meters tall. So let's draw that. So it's cone-shaped, and it has... Read More

Key Insights

• ⌛ The volume of a cone can be calculated using the formula 1/3 times the area of the base times the height.
• ⚾ Similar triangles can be used to find the radius of the base of the cone.
• 🍱 The number of complete boxes that can be filled is determined by dividing the volume of grain by the volume of each box.

Questions & Answers

Q: What is the volume of grain in the hopper?

The volume of grain in the hopper can be calculated by multiplying 1/3, the area of the base (pi times the radius squared), and the height. In this case, it is approximately 353.4 cubic meters.

Q: How many complete boxes will the grain fill?

Each box has a volume of 0.100 cubic meters. By dividing the volume of grain by a tenth of a cubic meter, we find that approximately 3,534 boxes can be filled.

Q: How is the volume of each box calculated?

The volume of each box is found by multiplying the width, depth, and height of the box together. In this case, it is 0.5 meters times 0.5 meters times 0.4 meters, equal to 0.100 cubic meters or a tenth of a cubic meter.

Q: How long does it take to fill the boxes?

To find this out, we divide the total volume of grain (353.4 cubic meters) by the rate at which it is poured (8 cubic meters per minute). The result is approximately 44 minutes to fill all of the boxes.

Summary & Key Takeaways

• A cone-shaped grain hopper has a radius of 10 meters at the top and is 8 meters tall. It is filled up to 2 meters from the top with grain.

• The volume of grain in the hopper is calculated using the formula for the volume of a cone: 1/3 times the area of the base times the height.

• The grain is poured into boxes with dimensions of 0.5 meters by 0.5 meters by 0.4 meters.

• The number of complete boxes that can be filled is calculated by dividing the volume of grain by the volume of each box.

• The time it takes to fill the boxes is calculated by dividing the total volume of grain by the rate at which it is poured.