The polynomial V(x)=x^3+9x^2-16x-144 represents volume of a shipping crate. Explain how to find the area of the base of the crate? if the polynomial H(x)=x+4 represents the height of the crate

x = − 9 , − 4 , 4

Step-by-step explanation:

Each crate is in the shape of a rectangular solid. Its dimensions are length, width, and height. The rectangular solid shown in the image below has a length  

4  units, width  2  units, and height  3

units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.

Breaking a rectangular solid into layers makes it easier to visualize the number of cubic units it contains. This  

4  by  2  by  3  rectangular solid has  24 cubic units.

A rectangular solid is shown. Each layer is composed of 8 cubes, measuring 2 by 4. The top layer is pink. The middle layer is orange. The bottom layer is green. Beside this is an image of the top layer that says  

Altogether there are  24  cubic units. Notice that  24  is the  length × width × height .

The top line says V equals L times W times H. Beneath the V is 24, beneath the equal sign is another equal sign, beneath the L is a 4, beneath the W is a 2, beneath the H is a 3.

The volume,  V , of any rectangular solid is the product of the length, width, and height.

V = L W H

We could also write the formula for the volume of a rectangular solid in terms of the area of the base. The area of the base,  B , is equal to  length × width . B = L ⋅ W

We can substitute  

B  for  L ⋅ W  in the volume, formula to get another form of the volume formula.

The top line says V equals red L times red W times H. Below this is V equals red parentheses L times W times H. Below this is V equals red capital B times h.

We now have another version of the volume formula for rectangular solids. Let’s see how this works with the  

4 × 2 × 3

rectangular solid we started with. See the image below.

 Besides the solid is V equals Bh. Below this is V equals Base times height. Below Base is parentheses 4 times 2. The next line says V equals parentheses 4 times 2 times 3. Below that is V equals 8 times 3, then V equals 24 cubic units.

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