Question

1.

A factory stacks boxes according to their weights. To provide stability to the stacks of boxes, heavier boxes are placed at the bottom of the stack, and lighter boxes are placed at the top. However, because of the boxes’ material, a box can only be placed on top of another if its weight is exactly half the weight of the lower box. Also, due to the height of the warehouse ceiling, boxes can only be stacked 4 levels high. The factory director has asked for your help in answering some questions about these boxes.

a. Explain how you would find the weight of each stacked box if you knew the weight of the bottom box. Find the weight of each box in a stack of 4 boxes if the bottom box weighs 10 pounds.
b. Eventually, these stacks of boxes will go onto pallets for shipping. We need to know how much each stack weighs in order to know how many stacks we can put on each pallet, but the bottom boxes do not necessarily weigh 10 pounds. In fact, we don’t know how much they weigh at all! Write and simplify an expression to find the weight of one stack of 4 boxes based on the unknown weight of the bottom box.
c. Each stack needs to weigh less than 100 pounds. Write and solve an inequality to find the maximum weight of the bottom box. What would be the possible range of weights for this box? It may help you to consider a graph of the solution to your inequality.
I rlly need help for c, I got the other 2.

Answer 1

If the weight of the bottom box is b then the weight of a stack of four boxes is b + b/2 + b/4 + b/8
= 8b/8 + 4b/8 + 2b/8 + 1b/8
= 15/8 b
A stack must weigh less than 100 pounds
15/8 b < 100
b < 100 x 8/15 = 53 1/3 pounds
If boxes have a minimum weight of 1 pound then
for stacks that contain 4 boxes the range of b is [8, 53 1/3)
and if stacks contain 1 to 4 boxes the range of b is [1, 53 1/3)

If the only restriction is a maximum of 100 then the range of b is [0, 53 1/3)