Find the dimensions of the box described. The length is 4 inches more than the width. The width is 2 inches more than the height. The volume is 135 cubic inches.

Question

This problem has been driving me insane. I cannot figure out how to solve it after multiple attempts, and I even asked my teacher about it and he didn't want to help. I have asked others too and they also have trouble with this problem. I am losing it...

Answer 1

The quantity that you know the LEAST ABOUT is the one where you assign the variable.

Read it again, which one do we know the least about...

Length is 4 inches more than the width... so we are told something about the length.

Width is 2 inches more than the height... so we are told something about eh width.

We know nothing about the HEIGHT, so let H be the height of the box

Width W = h+2

Length L = w + 4 = (h+2)+4 = h + 6

Volume = L * W * H

135 = (h+6)(h+2)h

135 = (h^2 + 8h + 12)h

The master equation becomes:

h^3 + 8h^2 + 12h - 135 = 0

Now for the rational root test.

p = {Factors of 135} = {+-1, +-3, +-5, +-9, +-15, +-27, +- 45, +-135}

q = {Factors of 1} = {+-1}

So p/q = p

The one that works is 3 : 3^3 + 8*3^2 + 12(3) - 135 = 27 + 72 + 36 - 135 = 0

So h=3 is a rational solution which means h-3 is a factor.

Synthetic division says:

_________________

3 | 1 8 12 -135

3 33 135

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1 11 45 0

So the quadratic that remains after (h-3) is factored is h^2 + 11h + 45

which does not factor and has imaginary/complex solutions

So h=3 is the only height available which means

the width is 5 and the length is 9

Answer 2

Hi Gunther,

I just read your question. I don't want to just give you the answer, as that doesn't help you learn anything, but I'll explain how to set up the solving method, and the fundamental thing to remember about Mathematical values getting multiplied.

So, remember that calculating volume involves multiplying the (3) individual values for the (3) dimensions in the box. In this question, your box is defined as having a length, width and height. So, the calculation for the volume would be: (length) x (width) x (height) = 135.

In the question, the dimensions are defined in relation to each other. One statement reads "the width is 2 inches more than the height." Mathematically, using variables, you could express height as "H" and width as "H + 2." The other statement in the question reads "the length is 4 inches more than the width." Since width and height are both expressed in terms of "H," use "H" when defining the length. With the length being 4 inches more than the width, you can define the length as "H + 2" + 4, or "H + 6."

So, with volume being (length) x (width) x (height) = 135, your equation would be:

(H + 6) x (H + 2) x (H) = 135

Finally, remember that when you multiply 3 values and the answer is odd, you can have (2) scenarios: only one of the values is odd, or all 3 values are odd. I would solve this by trial and error. Pick a value for "H" and solve from there. Good luck!