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The interior of a rectangular carton is designed by a certain manufacturer to have a volume of \(x\) cubic feet and a

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The interior of a rectangular carton is designed by a certain manufacturer to have a volume of \(x\) cubic feet and a

by VJesus12 » Tue Sep 15, 2020 5:27 am

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The interior of a rectangular carton is designed by a certain manufacturer to have a volume of \(x\) cubic feet and a ratio of length to width to height of \(3:2:2.\) In terms of \(x,\) which of the following equals the height of the carton, in feet?

A. \(\sqrt[3]{x}\)

B. \(\sqrt[3]{\dfrac{2x}3}\)

C. \(\sqrt[3]{\dfrac{3x}2}\)

D. \(\dfrac23\sqrt[3]{x}\)

E. \(\dfrac32\sqrt[3]{x}\)

Answer: B

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Re: The interior of a rectangular carton is designed by a certain manufacturer to have a volume of \(x\) cubic feet and

by deloitte247 » Sat Sep 19, 2020 12:34 pm
Given that length : width : height => 3n : 2n : 2n
Same positive integers n with respect to the whole perimeter
volume = x cubic ft
$$x=length*width*height$$
$$x=3n*2n*2n$$
$$x=12n^3$$
$$Make\ n\ subject\ of\ formular$$
$$\frac{12n^3}{12}=\frac{x}{12}$$
$$3\sqrt{n^3}=3\sqrt{\frac{x}{12}}$$
$$n=3\sqrt{\frac{x}{12}}$$
$$from\ the\ given\ ratios,\ height\ =\ 20\ where\ n=3\sqrt{\frac{x}{12}}$$
$$Therefore,\ height\ =2\cdot3\sqrt{\frac{x}{12}}$$
$$\exp res\sin g\ 2\ as\ a\ cubic\ root$$
$$height\ =3\sqrt{8}\cdot3\sqrt{\frac{x}{12}}$$
$$height\ =3\sqrt{\frac{8^2}{1}\cdot\frac{x}{12}}$$
$$height\ =3\sqrt{\frac{2}{1}\cdot\frac{x}{3}}$$
$$height\ =3\sqrt{\frac{2x}{3}}$$
$$Answer\ =\ B$$
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