A right rectangular prism has a height of 9 inches. The length of the prism's base is x inches, which is 7 inches more than the width of the prism's base. Which function V gives th... A right rectangular prism has a height of 9 inches. The length of the prism's base is x inches, which is 7 inches more than the width of the prism's base. Which function V gives the volume of the prism, in cubic inches, in terms of the length of the prism's base?
Understand the Problem
The question is asking for the volume function of a right rectangular prism given the relationships between its dimensions. The height is fixed at 9 inches, the length is expressed in terms of a variable x, and the width is given as x + 7. We need to construct the volume formula using these dimensions.
Answer
The function for the volume of the prism is \( V(x) = 9x(x + 7) \).
Answer for screen readers
The volume function is given by ( V(x) = 9x(x + 7) ).
Steps to Solve
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Identify the dimensions of the prism The problem states the height of the prism is fixed at 9 inches. The length of the base is given as $x$ inches, and the width of the prism's base is $x + 7$ inches.
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Write the volume formula for a rectangular prism The volume $V$ of a right rectangular prism can be calculated using the formula: $$ V = \text{length} \times \text{width} \times \text{height} $$
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Substitute the values into the volume formula Replace the length, width, and height in the volume formula:
- Length = $x$
- Width = $x + 7$
- Height = $9$
So the formula becomes: $$ V(x) = x \cdot (x + 7) \cdot 9 $$
- Simplify the volume expression Now, simplify the expression: $$ V(x) = 9x(x + 7) $$
This can also be expressed as: $$ V(x) = 9x^2 + 63x $$
The volume function is given by ( V(x) = 9x(x + 7) ).
More Information
This means that the volume of the prism, in terms of the length of the base which is ( x ), can be expressed as ( V(x) = 9x^2 + 63x ). This formula shows how the volume changes depending on the value of ( x ).
Tips
- Forgetting to account for the height when calculating the volume.
- Confusing the width and length dimensions.
- Misinterpreting the relationship between the width and length (e.g., adding instead of recognizing that width is ( x + 7 )).
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