Question

Alex wants to make an open top box from a flat rectangular piece of cardboard measuring 6 inches by 10 inches. To do this Alex will cut identical

squares out of the corners of the cardboard and then fold up the sides. Alex wants to use this box to store some of his card collection. He calculated
that he will need the area of the base of the box to be 34 square inches to fit all the cards he has in mind. What is the length of the sides of the squares
Alex needs to cut out of his original cardboard to get the size of box he wants?
• Write your answer as a number only.
• Round your answer to 2 decimal places if necessary.
Your Answer

Answer 1

To create the open-top box, Alex needs to remove squares of the same size from each corner of the rectangular cardboard piece. Let's assume that the side length of each square to be cut out is "x".

Then the dimensions of the cardboard after the squares are cut out would be:

Length: 10 inches minus 2x (as 2 squares are removed from the length)

Width: 6 inches minus 2x (as 2 squares are removed from the width)

After folding up the sides, the height of the box would also be "x".

The area of the base of the box is the product of the length and width of the cardboard piece after the squares have been cut out:

(10 - 2x)(6 - 2x) = 34

Expanding the left side of the equation and simplifying, we get:

4x² - 32x + 56 = 0

Dividing both sides by 4, we get:

x² - 8x + 14 = 0

Solving this quadratic equation using the quadratic formula or factoring, we get:

x = 2 or x = 6

Since the length and width of the cardboard are 10 and 6 inches respectively, the value of x cannot be greater than 3 (otherwise, the squares to be cut out will be larger than the length/width of the cardboard). Therefore, the only possible solution is:

x = 2 inches

Therefore, the length of the sides of the squares Alex needs to cut out of his original cardboard is 2 inches.