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# The length, width, and height of a rectangular box are in the ratio of

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Math Expert
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The length, width, and height of a rectangular box are in the ratio of [#permalink]

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29 Mar 2018, 00:09
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The length, width, and height of a rectangular box are in the ratio of 6:4:5, respectively. Express the height of the box in terms of its volume V.

A. $$\frac{1}{2}*\sqrt[3]{\frac{V}{3}}$$

B. $$\frac{5}{6}*\sqrt[3]{\frac{V}{15}}$$

C. $$2*\sqrt[3]{\frac{V}{15}}$$

D. $$\frac{5}{2}*\sqrt[3]{\frac{V}{15}}$$

E. $$3*\sqrt[3]{\frac{V}{15}}$$
[Reveal] Spoiler: OA

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Re: The length, width, and height of a rectangular box are in the ratio of [#permalink]

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29 Mar 2018, 01:00
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Expert's post
Bunuel wrote:
The length, width, and height of a rectangular box are in the ratio of 6:4:5, respectively. Express the height of the box in terms of its volume V.

A. $$\frac{1}{2}*\sqrt[3]{\frac{V}{3}}$$

B. $$\frac{5}{6}*\sqrt[3]{\frac{V}{15}}$$

C. $$2*\sqrt[3]{\frac{V}{15}}$$

D. $$\frac{5}{2}*\sqrt[3]{\frac{V}{15}}$$

E. $$3*\sqrt[3]{\frac{V}{15}}$$

As the calculation we're expected to do is straightforward, we'll just do it.
This is a Precise approach.

Writing V for volume and x for the height, we have:
$$V = x *( \frac{x}{5}*4) * (\frac{x}{5}*6) = \frac{24x^3}{25}$$
(Since the width is $$\frac{x}{5}*4$$and the length is $$\frac{x}{5}*6$$)

Simplifying gives
$$x = \sqrt[3]{\frac{25V}{24}}=\sqrt[3]{\frac{5^{2}V}{3*2^{3}}}=\sqrt[3]{\frac{5^3V}{5*3*2^3}}=\frac{5}{2}*\sqrt[3]{\frac{V}{15}}$$

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Re: The length, width, and height of a rectangular box are in the ratio of [#permalink]

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30 Mar 2018, 10:55
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Expert's post
Bunuel wrote:
The length, width, and height of a rectangular box are in the ratio of 6:4:5, respectively. Express the height of the box in terms of its volume V.

A. $$\frac{1}{2}*\sqrt[3]{\frac{V}{3}}$$

B. $$\frac{5}{6}*\sqrt[3]{\frac{V}{15}}$$

C. $$2*\sqrt[3]{\frac{V}{15}}$$

D. $$\frac{5}{2}*\sqrt[3]{\frac{V}{15}}$$

E. $$3*\sqrt[3]{\frac{V}{15}}$$

We can create the ratio of length : width : height = 6x : 4x : 5x

The volume is 6x * 4x * 5x = 120x^3. So:

120x^3 = V

x^3 = V/120

x = ∛(V/120)

x = ∛(V/(8*15))

x = ∛(⅛) ∛(V/15)

x = ½ ∛(V/15)

Since the height is 5x, height = 5 * ½ ∛(V/15) = 5/2 ∛(V/15).

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Re: The length, width, and height of a rectangular box are in the ratio of   [#permalink] 30 Mar 2018, 10:55
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