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Advanced Math / Nonlinear functions Difficulty: Hard

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?

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Explanation

Choice D is correct. The volume of a right rectangular prism can be represented by a function $V$ that gives the volume of the prism, in cubic inches, in terms of the length of the prism's base. The volume of a right rectangular prism is equal to the area of its base times its height. It's given that the length of the prism's base is $x$ inches, which is $7$ inches more than the width of the prism's base. This means that the width of the prism's base is $x minus 7$ inches. It follows that the area of the prism's base, in square inches, is $x left parenthesis x minus 7 right parenthesis$ and the volume, in cubic inches, of the prism is $x left parenthesis x minus 7 right parenthesis left parenthesis 9 right parenthesis$ . Thus, the function $V$ that gives the volume of this right rectangular prism, in cubic inches, in terms of the length of the prism's base, $x$ , is $V (x) = 9 x (x - 7)$ .

Choice A is incorrect. This function would give the volume of the prism if the height were $9$ inches more than the length of its base and the width of the base were $7$ inches more than its length.

Choice B is incorrect. This function would give the volume of the prism if the height were $9$ inches more than the length of its base.

Choice C is incorrect. This function would give the volume of the prism if the width of the base were $7$ inches more than its length, rather than the length of the base being $7$ inches more than its width.