Jan has a 18X18X12 inch gift box that needs to be placed carefully into a 2X2X2ft shipping carton, surrounded by packing peanuts.
a. How many 1-cubic-foot bags of peanuts does Jan need to buy?
b. Jan opens one bag of peanuts and spreads them evenly on the bottom of the shipping carton. What is the resulting depth of the peanuts?
c. Jan centers the square base of the gift box on the peanut layer,pours in another bag of peanuts, and spreads them evenly around. Now how deep are the peanuts?
d. Explain why the third bag of peanuts will cover the gift box.
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a. To determine how many 1-cubic-foot bags of peanuts Jan needs to buy, we need to find the total volume of the shipping carton.
The shipping carton has dimensions of 2ft x 2ft x 2ft, which can be converted to inches as:
2ft = 2 * 12 inches = 24 inches
The volume of the shipping carton is calculated by multiplying the length, width, and height:
Volume = (24 inches) * (24 inches) * (24 inches)
Volume = 13,824 cubic inches
Since each 1-cubic-foot bag contains 12 * 12 * 12 = 1,728 cubic inches, we can divide the volume of the shipping carton by the volume of each bag to find the number of bags needed:
Number of bags = 13,824 cubic inches / 1,728 cubic inches
Number of bags = 8 bags
Therefore, Jan needs to buy 8 bags of peanuts.
b. After opening one bag of peanuts and spreading them evenly on the bottom of the shipping carton, the resulting depth of the peanuts would be equal to the height of the peanuts in the bag. Since each bag is 1 cubic foot, the depth of the peanuts is 1 foot or 12 inches.
c. After centering the square base of the gift box on the bottom peanut layer, pouring in another bag of peanuts, and spreading them evenly around, the depth of the peanuts would increase. Adding another bag of peanuts, which is 1 cubic foot or 12 inches in depth, would result in a total depth of 12 inches + 12 inches = 24 inches.
d. The third bag of peanuts will cover the gift box because the total depth of the peanuts after adding the second bag is 24 inches, which is higher than the height of the gift box, which is 12 inches. Thus, the third bag of peanuts will have enough depth to fully cover the gift box.
a. To find out how many 1-cubic-foot bags of peanuts Jan needs to buy, we need to calculate the volume of the gift box, and then divide it by the volume of each bag of peanuts.
The volume of the gift box can be found by multiplying its length, width, and height. In this case, the gift box has dimensions of 18 x 18 x 12 inches. However, we need to convert these inches to feet to match the units of the shipping carton. Since there are 12 inches in a foot, the dimensions of the gift box can be converted to 18/12 x 18/12 x 12/12 feet, which is equal to 1.5 x 1.5 x 1 feet.
The volume of the gift box is therefore 1.5 x 1.5 x 1 = 2.25 cubic feet.
Now, we need to divide the volume of the gift box by the volume of each bag of peanuts (which is 1 cubic foot).
Number of bags of peanuts = Volume of gift box / Volume of each bag
Number of bags of peanuts = 2.25 cubic feet / 1 cubic foot = 2.25 bags
Since we cannot buy a fraction of a bag, Jan would need to buy 3 bags of peanuts.
b. When Jan opens one bag of peanuts and spreads them evenly on the bottom of the shipping carton, the resulting depth of the peanuts would be equal to the height of the bag, which is 1 foot.
c. After Jan centers the square base of the gift box on the peanut layer, pours in another bag of peanuts, and spreads them evenly around, the total depth of the peanuts would be the sum of the initial depth (1 foot) and the height of the second bag (1 foot). Therefore, the peanuts would be 2 feet deep.
d. The third bag of peanuts will cover the gift box because the total depth of the peanuts is already 2 feet, which is equal to the height of the shipping carton. Therefore, the gift box would be completely covered by the third bag of peanuts.