Question

A shipping company is creating their own shipping boxes. They have created a box with a volume of 800in^3 , a height of 8 inches and a base perimeter of 40 inches.

The machine malfunctioned, and in order to be fixed the correct dimensions of the box needed to be input. What length and width would a worker need to input in order to fix the machine?


Length:

Width:

Answer 1

Answer: The worker would need to input Length: 10 inches and Width: 10 inches

Step-by-step explanation: The boxes as described in the question have a volume of 800 cubic inches. The height is given as 8 inches and the base length and width is not given, however the perimeter is given as 40 inches. This gives us a clue as to the dimensions of the base length and width as follows;

Perimeter = 2(L + W)

40 = 2(L + W)

40/2 = L + W

20 = L + W ---------- (1)

This means the addition of the length and width gives us a total of 20 inches.

However, note that the volume of the box is derived as;

Volume = L x W x H

With the volume and height already given as 800 and 8 respectively, the formula becomes;

800 = L x W x 8

Divide both sides of the equation by 8

100 = L x W ----------(2)

We can now solve for the pair of simultaneous equations as follows;

20 = L + W ----------(1)

100 = L x W ----------(2)

From equation (1), L = 20 - W

Substitute for the value of L into equation (2)

100 = (20 - W) * W

100 = 20W - W²

Collect like terms and you now have

W² -20W + 100 = 0

By factorization we can solve the above quadratic equation as;

(W - 10W) ( W - 10W) = 0

W - 10 = 0

W = 10

From equation (1), when W = 10, then

20 = L + 10

Subtract 10 from both sides of the equation

10 = L

Therefore, the worker would have to input, Length : 10 inches, and Width : 10 inches

Answer 2

Length = 10 inches

Width = 10 inches

Explanation:

The perimeter of the base is 40 inches, so we can write the following equation:

2*Length + 2*Width = 40

Length + Width = 20 (eq1)

The volume of the box is 800 in3, so we have:

800 = Length*Width*8

Length*Width = 100 (eq2)

From the (eq1), we have:

Length = 20 - Width

Using this value of Length in (eq2), we have:

(20 - Width)*Width = 100

20*Width - Width^2 - 100 = 0

Using Bhaskara's formula, we have:

Delta = b^2 - 4ac = 400 - 400 = 0

Width = -b/2a = -20/(-2) = 10 inches

Length = 20 - Width = 20 - 10 = 10 inches

So the length is 10 inches and the width is 10 inches