Please answer this word problem! With solutions and explanation would be very much appreciated. Thank you!

Question

The height of a box is twice its length. The length of this box is twice its width. If each dimension is increased by 1 inch, the volume is increased by 71 square inches. Find the dimensions of the original box.

Answer 1

As with all word problems, once you translate the sentences into equations it becomes quite straightforward. Let's go through each, letting h be the height, l the length, w the width, and V the volume.

 

The height of a box is twice it's length: h = 2l

The length of the box is twice its width: l = 2w

note that you may substitute the second equation into the first to get: h = 2(2w) = 4w

 

And finally the harder sentence. (note: I assume you meant the volume is increased by 71 cubic inches). We will use V as the box's original volume. You should already know that the volume of a box is described by the equation V = lwh. To transform this sentence to an equation, simply add 1 to each dimension and 71 to the total volume:

 

(l+1)(w+1)(h+1) = V+71 = lwh+71

 

Now with three equations and three unknowns, you should be able to carry the problem through on your own. However, as you requested a full solution, I will continue. Let's solve with substitution, using the first two equations to get everything in terms of w.

 

(2w+1)(w+1)(4w+1) = (2w)(w)(4w)+71 = 8w³+71

 

multiply out the left side to get:

 

8w³+14w²+7w+1 = 8w³+71

 

subtract from both sides to get:

 

14w²+7w-70 = 0

 

Factor:

 

7(2w²+w-10) = 7(w-2)(2w+5) = 0

 

note that for that equation to be true, either w-2 = 0 or 2w+5 = 0. Solving those two equations gives us w = 2 or w = -5/2. We can safely throw out the negative answer, as in our original context it won't make sense. Now we simply use w = 2 in our first two equations to get our full answer:

 

w = 2

l = 2w = 2(2) = 4

h = 2l = 2(4) = 8

 

And as always lets check to make sure this is correct by calculating the volume:

 

V = lwh = (4)(2)(8) = 64

and

(l+1)(w+1)(h+1) = (5)(3)(9) = 135 = 64+71

 

So everything checks out!

 

Answer 2

We know the volume of a box is length times width times height.

 

Let width = x

Let length = 2x

Let height = 2(2x) = 4x

 

These variables represent the dimensions of the original box.  When we multiply them out, the original volume is 8x³.

 

But wait.  When we increase the size of the box, we also increase the volume.  When each original dimension is increased by one, the volume is increase by 71.  We can write an equation to represent this.

 

(x + 1)(2x + 1)(4x + 1) = 8x³ + 71

 

(x + 1)(8x² + 6x + 1) = 8x³ + 71

 

x(8x² + 6x + 1) + (8x² + 6x + 1) = 8x³ + 71

 

8x³ + 14x² + 7x + 1 = 8x³ + 71

 

 

Make the right side equal to zero by subtracting 8x³ and subtracting 71 on both sides of the equation.

 

14x² + 7x - 70 = 0

 

 

Factor out a 7.

 

7(2x² + x - 10) = 0

 

 

Set the term in parenthesis equal to zero, since any number times zero is always zero.

 

2x² + x - 10 = 0

 

(2x + 5)(x - 2) = 0

 

2x + 5 = 0           and          x - 2 = 0

 

x = -5/2               and          x = 2

 

 

Since we cannot have a negative value for dimensions, we accept the positive value of x:

 

x = 2

 

 

Now, we plug in this value of x into the dimensions of the original box.

 

width = 2 in

length = 4 in

height = 8 in

 

 

To check, we multiply these dimensions to see if we get the original volume of 8x³.

 

2 * 4 * 8 = 8(2)³

 

64 = 64

 

 

The original volume is 64 in³.   Now we use this volume to check the volume increase.

 

(2 + 1)(4 + 1)(8 + 1) = 64 + 71

 

3 * 5 * 9 = 135

 

135 = 135

 

 

It all checks out!!

 

 

 

 

 

Answer 3

Hello Alex,

First, write the system of equation pertinent to the problem definition assuming L, W, H stand for the length, width and height:

 

L=2W
H=2L
(W+1)(L+1)(H+1)=LWH+71

Notice that: H=4W and write:

(W+1)(2W+1)(4W+1)=W(2W)(4W)+71
(W+1)(8W²+2W+4W+1)=8W³+71
(W+1)(8W²+6W+1)=8W³+71
8W³+6W²+W+8W²+6W+1=8W³+71

Resulting in Quadratic Equation (you can solve it using online Quadratic Equation Solver http://examn8.com/QuadraticEquations.aspx):

14W²+7W-70=0

which has two solutions but only one positive:

W=2 (width)
Correspondingly:
L=4 (length)
H=8 (height)

Hope this will help.