Question

(a) the rectangle and the circle below have the same area. what is the area?
(π m, 9 m)
area of rectangle = □ m²
(radius is 3 m)
area of circle = □ m²
(3 m)
(b) the rectangle in part (a) is a cross section parallel to the base of the prism below. the height of the prism is 8 m. find the volume of the prism.
(π m, 9 m, 8 m)
volume of prism = □ m³

Explanation:

Step1: Solve part (a) - Area of Rectangle

The area of a rectangle is given by the formula \( A = \text{length} \times \text{width} \). From the diagram, the length of the rectangle is \( 9 \, \text{m} \) and the width is \( \pi \, \text{m} \)? Wait, no, actually, since the rectangle and the circle have the same area, we can first find the area of the circle and that will be the area of the rectangle. Wait, the circle has a radius \( r = 3 \, \text{m} \). The area of a circle is \( A = \pi r^2 \). So let's calculate the area of the circle first.

Wait, maybe I misread. Let's check again. The problem says "The rectangle and the circle below have the same area." So first, find the area of the circle, then that's the area of the rectangle.

Step1: Area of Circle

The formula for the area of a circle is \( A = \pi r^2 \), where \( r = 3 \, \text{m} \).
So \( A = \pi \times (3)^2 = 9\pi \, \text{m}^2 \). So the area of the rectangle is also \( 9\pi \, \text{m}^2 \). Wait, but let's check the rectangle's dimensions. The rectangle has length \( 9 \, \text{m} \) and width \( \pi \, \text{m} \)? Wait, \( 9 \times \pi = 9\pi \), which matches the circle's area. So that's consistent.

Step2: Area of Circle (to confirm)

Wait, maybe the problem is to find the area of the rectangle and the area of the circle, and they should be equal. So:

For the rectangle: length \( l = 9 \, \text{m} \), width \( w = \pi \, \text{m} \)? Wait, no, the diagram shows the rectangle with height \( 9 \, \text{m} \) and width \( \pi \, \text{m} \)? Wait, maybe the width is \( \pi \, \text{m} \) and length \( 9 \, \text{m} \). So area of rectangle is \( 9 \times \pi = 9\pi \, \text{m}^2 \).

For the circle: radius \( r = 3 \, \text{m} \), area is \( \pi r^2 = \pi \times 3^2 = 9\pi \, \text{m}^2 \). So both areas are \( 9\pi \, \text{m}^2 \).

Step3: Solve part (b) - Volume of Prism

The volume of a prism is given by the formula \( V = \text{Area of cross - section} \times \text{length (or height, depending on orientation)} \). The cross - section is the rectangle from part (a), which has area \( 9\pi \, \text{m}^2 \), and the height of the prism is \( 8 \, \text{m} \) (wait, the diagram shows the prism with a dimension of \( 8 \, \text{m} \) and \( 9 \, \text{m} \)? Wait, the prism is a rectangular prism? Wait, the cross - section is the rectangle from part (a), and the height of the prism (the distance between the cross - sections) is \( 8 \, \text{m} \)? Wait, the diagram shows the prism with a length of \( 8 \, \text{m} \), and the cross - section (the rectangle) has area \( 9\pi \, \text{m}^2 \). Wait, no, maybe the cross - section is the rectangle with area \( 9\pi \, \text{m}^2 \), and the length of the prism (the distance along the prism) is \( 8 \, \text{m} \). Wait, the formula for the volume of a prism is \( V=\text{Area of base (cross - section)} \times \text{height (length of the prism)} \).

Wait, the problem says "The rectangle in part (a) is a cross - section parallel to the base of the prism. The height of the prism is \( 8 \, \text{m} \). Find the volume of the prism."

So the area of the cross - section (which is the same as the area of the rectangle, which is \( 9\pi \, \text{m}^2 \)) and the height of the prism is \( 8 \, \text{m} \)? Wait, no, looking at the prism diagram, there's a dimension of \( 8 \, \text{m} \) and \( 9 \, \text{m} \)? Wait, maybe I misread. Wait, the prism has a cross - section (the rectangle) with area \( 9\pi \, \text{m}^2 \), and the length of the prism (the distance perpendicular to the cross - section) is \( 8 \, \text{m} \)? Wait, no, the diagram of the prism shows a dimension of \( 8 \, \text{m} \) and \( 9 \, \text{m} \)? Wait, maybe the cross - section is the rectangle with area \( 9\pi \, \text{m}^2 \), and the length of the prism is \( 8 \, \text{m} \). Wait, no, let's check the formula for the volume of a prism: \( V = \text{Area of base} \times \text{height} \), where the height is the distance between the two bases (cross - sections).

Wait, maybe the cross - section (the rectangle) has area \( 9\pi \, \text{m}^2 \), and the length of the prism (the side perpendicular to the cross - section) is \( 8 \, \text{m} \). Wait, no, the diagram of the prism shows a dimension of \( 8 \, \text{m} \) and \( 9 \, \text{m} \)? Wait, maybe I made a mistake in part (a). Let's re - do part (a) correctly.

Part (a)

- **Area of Rectangle**: The rectangle has length \( 9 \, \text{m} \) and width \( \pi \, \text{m} \)? Wait, no, the diagram shows the rectangle with height \( 9 \, \text{m} \) and width \( \pi \, \text{m} \). So area \( A = \text{length} \times \text{width} = 9 \times \pi = 9\pi \, \text{m}^2 \).
- **Area of Circle**: The circle has radius \( r = 3 \, \text{m} \). Area \( A=\pi r^2=\pi\times3^2 = 9\pi \, \text{m}^2 \). So both areas are \( 9\pi \, \text{m}^2 \).

Part (b)

The volume of a prism is given by \( V=\text{Area of cross - section}\times\text{length of the prism} \). The cross - section is the rectangle from part (a) with area \( 9\pi \, \text{m}^2 \). Wait, but looking at the prism diagram, there's a dimension of \( 8 \, \text{m} \). Wait, maybe the cross - section is actually a rectangle with length \( 8 \, \text{m} \) and width \( 9 \, \text{m} \)? No, the problem says "The rectangle in part (a) is a cross - section parallel to the base of the prism." So the cross - section area is the area of the rectangle from part (a), which is \( 9\pi \, \text{m}^2 \), and the height of the prism (the distance between the cross - sections) is \( 8 \, \text{m} \)? Wait, no, maybe I misread the prism's dimensions. Wait, the prism diagram shows a dimension of \( 8 \, \text{m} \) (maybe the length) and the cross - section is the rectangle with area \( 9\pi \, \text{m}^2 \). Wait, no, let's check the formula again. The volume of a rectangular prism is \( V = l\times w\times h \), but if the cross - section is the rectangle from part (a), then the cross - section area is \( A = 9\pi \, \text{m}^2 \), and the length of the prism (the side perpendicular to the cross - section) is \( 8 \, \text{m} \). Wait, no, the diagram of the prism has a dimension of \( 8 \, \text{m} \) and \( 9 \, \text{m} \), but maybe the cross - section is the rectangle with area \( 9\pi \, \text{m}^2 \), and the height of the prism is \( 8 \, \text{m} \). Wait, I think I made a mistake. Let's re - examine the problem.

Wait, the problem says: "The rectangle in part (a) is a cross - section parallel to the base of the prism. The height of the prism is \( 8 \, \text{m} \). Find the volume of the prism."

The volume of a prism is \( V=\text{Area of cross - section}\times\text{height of the prism} \). The cross - section (the rectangle from part (a)) has area \( 9\pi \, \text{m}^2 \), and the height of the prism is \( 8 \, \text{m} \)? Wait, no, maybe the cross - section is a rectangle with length \( 8 \, \text{m} \) and width \( 9 \, \text{m} \), but no, part (a) says the rectangle and circle have the same area. Wait, maybe the cross - section is the rectangle with area \( 9\pi \, \text{m}^2 \), and the length of the prism (the distance along the prism) is \( 8 \, \text{m} \). So \( V = 9\pi\times8 = 72\pi \, \text{m}^3 \)? Wait, no, that doesn't seem right. Wait, maybe the prism is a rectangular prism with length \( 8 \, \text{m} \), width \( \pi \, \text{m} \), and height \( 9 \, \text{m} \)? Then volume would be \( 8\times\pi\times9 = 72\pi \, \text{m}^3 \). Yes, that makes sense. Because the cross - section parallel to the base (which is a rectangle) would have area \( \pi\times9 = 9\pi \) (if the base is \( \pi \times 9 \)) and the length of the prism is \( 8 \, \text{m} \), so volume is \( 9\pi\times8 = 72\pi \).

Wait, let's clarify:

- For part (a), area of rectangle: length \( 9 \, \text{m} \), width \( \pi \, \text{m} \), so \( A = 9\times\pi = 9\pi \, \text{m}^2 \).
- Area of circle: \( \pi r^2=\pi\times3^2 = 9\pi \, \text{m}^2 \), so they are equal.

- For part (b), the prism has a cross - section (the rectangle from part (a)) with area \( 9\pi \, \text{m}^2 \), and the height of the prism (the distance between the cross - sections) is \( 8 \, \text{m} \). So volume \( V=\text{Area of cross - section}\times\text{height}=9\pi\times8 = 72\pi \, \text{m}^3 \).

Answer:

(a) Area of rectangle: \( 9\pi \, \text{m}^2 \), Area of circle: \( 9\pi \, \text{m}^2 \)

(b) Volume of prism: \( 72\pi \, \text{m}^3 \)

Wait, but let's check the problem again. The problem (a) says "What is the area?" (for both rectangle and circle). So:

• Area of rectangle: \( 9\pi \, \text{m}^2 \)
• Area of circle: \( 9\pi \, \text{m}^2 \)

For (b), volume of prism: The formula for the volume of a prism is \( V = \text{Area of base} \times \text{height} \). The base is the rectangle from (a) with area \( 9\pi \, \text{m}^2 \), and the height of the prism is \( 8 \, \text{m} \), so \( V = 9\pi\times8 = 72\pi \, \text{m}^3 \).