Find the volume of the wood used to make the prism and the total surface area of the prism

Question

Find the volume of the wood used to make the prism and the total surface area of the prism

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Answer 1

Let's find the volume of the wood:

We can start by separating the piece of wood into two geometric shapes to make finding the volume easier. Let's separate them into a rectangular and a triangular prism, find the volume of each individual shape, then find the sum.

The rectangular prism now has dimensions of 3x4x9. To find the volume, we multiply to get 108 cm^3.

The triangular prism now has dimensions B=8, W = 4, and H = 6. To find the volume of a triangular prism, we do 1/2xBxWxH. This gives us 1/2x8x4x6 = 96 cm^3.

Now, we add these to find the volume of the entire piece of wood. 96 cm^3 + 108 cm^3 = 204 cm^3.

Now to find the surface area of the shape, we need to find the surface area of each face and add them together. To find the surface area of a rectangular face, we multiply BxH. For a tringular face, 1/2xBxH.

Let's start with the rectangular faces, of which there are 5.

Two of them share the same dimensions: 4x3. These each have Surface area (SA) =12 cm^2. The third face has dimension 9x4, which has a SA=36cm^2. The fourth rectangular face has dimensions 8x4 = 32cm^2. The last face has dimensions 7Xx4 = 28cm^2

These faces total up 12+12+36+32+28 = 120cm^2.

Now for the other 2 faces. They have the same dimensions, so we can find the SA of one face and double it to find the SA of both. Each face can be separated into a rectangle with dimensions 3x9 and a triangle with dimensions 5x6 to finding the SA easier. The SA of the rectangle is 9x3=27cm^2 each, and the SA of the triangle portion is 1/2x5x6 = 15 cm^2 each. If we add these together, we get 42cm^2 per face, with two faces = 84cm^2.

Lastly, we add the all together to get the final Surface area of the shape.

84 cm^2 + 120cm^2 = 204 cm^2

Answer 2

THE ANSWER IS: (a) 126cm² and (b) 129cm³

HERE IS A BREAKDOWN:

(a) wants us to find the VOLUME:

We can break the shape into two solid parts:

1. A rectangular prism
2. A triangular prism sitting beside it

For the Rectangular Prism, we use:

1. Length = 3 cm
2. Width = 4 cm
3. Height = 9 cm

The formula for the volume (V) of a rectangular prism is:

V= L x W x H so, V= 3 x 4 x 9

V= 108cm³

For the Triangular Prism, we use:

1. Triangle base (B) = 3 cm
2. Triangle height (H) = 3 cm
3. Depth (D) = 4 cm

First, find the area(A) of the triangle:

A= 1/2 x B x H so, A= 1/2 x 3 x 3

A= 4 and 1/2

To find the volume (V), we multiply by the depth (D):

V= 4 1/2 x 4 so, V=18cm³

The TOTAL Volume (V) can be found by adding each Volume (V) together:

108 + 18 = 126cm³

****SO the answer for part (a) is 126cm³

(b) wants us to find the total surface area

To do this, we find the area of each exposed face and add them all together.

Rectangular Faces (there are 5):

1. Left Side: 3 x 4 = 12cm²
2. Right side (behind triangle): 3 x 4 = 12cm²
3. Back face: 9 x 4 = 36cm²
4. Bottom: 8 x 4 = 32cm²
5. Slanted Top: 7 x 4 = 28cm²

Add each face together: 12 + 12 + 36 + 32 + 28 = 120cm²

Triangular Faces (there are 2):

In part (a), we found the area of the triangle to be 4 and 1/2 cm²

Since we have 2 sides with the SAME measurements, we can just double it.

2 x 4 1/2 = 9cm²

^{The TOTAL Surface Area can be found by adding these together:}

120 + 9 = 129cm²

****SO the answer for part (b) is 129cm²