A sheet of cardboard with an area of 10 square feet is made into a closed rectangular box. The four sides of the box have the same area and two ends have the same area. The area of each of the four equal sides is twice the area of each end. What is the area of each face of the box?
To solve this problem, we need to break it down into smaller steps:
Step 1: Let's represent the length and width of the cardboard sheet as 'L' and 'W' respectively, in feet.
Step 2: We are given that the area of the cardboard sheet is 10 square feet, so we can write the equation as L * W = 10.
Step 3: We are also given that the four sides of the box have the same area. Let's call this area 'S'. So, the total area of the four sides will be 4S.
Step 4: We are told that two ends of the box have the same area. Let's call this area 'E'. So, the total area of the two ends will be 2E.
Step 5: According to the problem, the area of each of the four equal sides is twice the area of each end. So, we can write the equation as S = 2E.
Step 6: Since the sheet of cardboard is folded to form a closed rectangular box, the total surface area of the box will be 2E + 4S.
Step 7: By substituting the values of S and E from equations (3) and (5) into equation (6), we can simplify the equation as follows:
Total Surface Area = 2E + 4S
= 2E + 4(2E) (substituting S = 2E)
= 2E + 8E
= 10E
Step 8: We are given that the total surface area is 10 square feet. So, we have the equation 10E = 10.
Step 9: Solving for E, we divide both sides of the equation by 10: E = 1.
Step 10: Now, we can substitute the value of E into the equation S = 2E to find the value of S: S = 2 * 1 = 2.
Step 11: Finally, we can calculate the area of each face of the box. We have two ends, each with an area of E = 1 square foot, and four sides, each with an area of S = 2 square feet.
Therefore, the area of each face of the box is 1 square foot for each end and 2 square feet for each side.