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

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

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New post 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. \(3*\sqrt[3]{\frac{V}{15}}\)

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

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New post 29 Mar 2018, 01:00
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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
\(x = \sqrt[3]{\frac{25V}{24}}=\sqrt[3]{\frac{5^{2}V}{3*2^{3}}}=\sqrt[3]{\frac{5^3V}{5*3*2^3}}=\frac{5}{2}*\sqrt[3]{\frac{V}{15}}\)

(D) is our answer.
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Re: The length, width, and height of a rectangular box are in the ratio of  [#permalink]

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New post 30 Mar 2018, 10:55
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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).

Answer: D
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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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