At a farm, animals are fed bales of hay and buckets of grain. Each bale of hay is in the shape a rectangular prism. The base has side lengths 2 feet and 3 feet, and the height is 5 feet. Each bucket of grain is a cylinder with a diameter of 3 feet. The height of the bucket is 5 feet, the same as the height of the bale.

Which is larger in area, the rectangular base of the bale or the circular base of the bucket? Explain how you know.

Which is larger in volume, the bale or the bucket? Explain how you know.

area of base of bale = 2(3) ft^2 =

area of circle = π(1.5)^2

evaluate then decide which has the bigger area

since both have the same height --> the larger area will yield the larger volume

To determine which base is larger in area, we need to compare the areas of the rectangular base of the bale and the circular base of the bucket.

The area of the rectangular base of the bale can be calculated by multiplying the lengths of its sides: 2 feet x 3 feet = 6 square feet.

The area of the circular base of the bucket can be calculated using the formula for the area of a circle, which is πr^2, where r is the radius of the circle. In this case, the diameter is given as 3 feet, so the radius is half of that, which is 1.5 feet. Therefore, the area of the circular base is π(1.5 feet)^2 ≈ 7.07 square feet.

Since the area of the circular base of the bucket is greater than the area of the rectangular base of the bale (7.07 square feet > 6 square feet), we can conclude that the circular base of the bucket is larger in area.

To determine which has a larger volume, we need to compare the volumes of the bale and the bucket.

The volume of the rectangular bale can be calculated by multiplying the lengths of its sides: 2 feet x 3 feet x 5 feet = 30 cubic feet.

The volume of the cylindrical bucket can be calculated using the formula for the volume of a cylinder, which is πr^2h, where r is the radius of the circular base and h is the height of the cylinder. In this case, the radius is 1.5 feet and the height is 5 feet. Therefore, the volume of the bucket is π(1.5 feet)^2 x 5 feet ≈ 35.34 cubic feet.

Since the volume of the bucket is greater than the volume of the bale (35.34 cubic feet > 30 cubic feet), we can conclude that the bucket has a larger volume.

To determine which is larger in area, the rectangular base of the bale or the circular base of the bucket, we need to calculate the areas of both shapes.

The area of the rectangular base of the bale can be found using the formula: Area = length × width. Given that the length is 2 feet and the width is 3 feet, we can calculate: Area of rectangular base = 2 feet × 3 feet = 6 square feet.

The area of the circular base of the bucket can be found using the formula: Area = π × radius^2. Given that the diameter is 3 feet, the radius is half of that, so the radius is 3 feet / 2 = 1.5 feet. Plugging this into the formula, we can calculate: Area of circular base = π × (1.5 feet)^2 = 7.07 square feet (approximately).

Comparing the two areas, we can see that the area of the circular base of the bucket is larger than the area of the rectangular base of the bale. Therefore, the circular base of the bucket is larger in area.

To determine which is larger in volume, the bale or the bucket, we need to calculate the volumes of both shapes.

The volume of the rectangular bale can be found using the formula: Volume = length × width × height. Given that the length is 2 feet, the width is 3 feet, and the height is 5 feet, we can calculate: Volume of bale = 2 feet × 3 feet × 5 feet = 30 cubic feet.

The volume of the cylindrical bucket can be found using the formula: Volume = π × radius^2 × height. Given that the radius is 1.5 feet and the height is 5 feet (same as the bale), we can calculate: Volume of bucket = π × (1.5 feet)^2 × 5 feet = 35.34 cubic feet (approximately).

Comparing the two volumes, we can see that the volume of the bucket is larger than the volume of the bale. Therefore, the bucket has a larger volume than the bale.

In summary:
- The circular base of the bucket is larger in area than the rectangular base of the bale.
- The bucket has a larger volume than the bale.