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The storage container below is in the shape of a rectangular prism with a height of 6 feet and a length that is 2 feet more than its width.

Recall that the formula for the volume of a rectangular prism is [tex]V = l \cdot w \cdot h[/tex], where [tex]l[/tex] is the length, [tex]w[/tex] is the width, and [tex]h[/tex] is the height.

Write the equation that represents the volume of the storage container in terms of its width.

A. [tex]V = 6w^2 + 12w[/tex]

B. [tex]V = 6w^2 - 12w[/tex]

C. [tex]V = 6w^2 - 12[/tex]

D. [tex]V = 6w^2 + 12[/tex]


Sagot :

Let's solve the problem step-by-step.

1. Define the variables:
- Let [tex]\( w \)[/tex] be the width of the rectangular prism.
- Let [tex]\( h \)[/tex] be the height of the rectangular prism, given as [tex]\( 6 \)[/tex] feet.
- Let [tex]\( l \)[/tex] be the length of the rectangular prism, which is given as [tex]\( 2 \)[/tex] feet more than its width, so [tex]\( l = w + 2 \)[/tex].

2. Recall the volume formula:
- The volume [tex]\( V \)[/tex] of a rectangular prism is given by the formula [tex]\( V = l \cdot w \cdot h \)[/tex].

3. Substitute the expressions for [tex]\( l \)[/tex] and [tex]\( h \)[/tex]:
- Length [tex]\( l = w + 2 \)[/tex]
- Height [tex]\( h = 6 \)[/tex] feet

This gives:
[tex]\[ V = (w + 2) \cdot w \cdot 6 \][/tex]

4. Rewrite the formula by distributing and combining like terms:
[tex]\[ V = 6w \cdot (w + 2) \][/tex]

5. Simplify the expression:
[tex]\[ V = 6w \cdot w + 6w \cdot 2 \][/tex]
[tex]\[ V = 6w^2 + 12w \][/tex]

Therefore, the equation that represents the volume of the storage container in terms of its width [tex]\( w \)[/tex] is:
[tex]\[ V = 6w^2 + 12w \][/tex]

The correct answer is:
A. [tex]\(\quad V = 6w^2 + 12w\)[/tex]