High School

5) A warehouse outside of a factory currently has an inventory of 1245 boxes. After an 8-hour work day, the warehouse has 2000 boxes. Assume the warehouse was being filled at a constant (linear) rate.

a) How many boxes per hour is the factory able to provide to the warehouse?

b) What would be the inventory at the end of a 40-hour work week?

c) How long will it take to fill the warehouse to its 50,000 box capacity?

6) In the year 2007, a FOREVER stamp cost $0.41. In 2023, the cost of a FOREVER stamp was $0.63. Assume that the cost of stamps increased at a constant (linear) rate.

a) If price increases continue at the current rate, how much will a FOREVER stamp cost in 2035?

b) In what year would you expect a FOREVER stamp to cost one dollar?

7) In January of 2021, there were 980,000 games available on the Apple App Store. By July of 2021, there were 984,200 games available. If we assume that the number of available games is steadily increasing at a constant (linear) rate,

a) How many games does this pattern predict will be available in January 2022?

b) At this rate, when will there be 1,000,000 games available for purchase in the Apple App Store?

Answer :

Thee factory is able to provide 94.38 boxes per hour to the warehouse.

How to calculate the value

Rate = (2000 - 1245) / 8 = 94.38 boxes per hour

Therefore, the factory is able to provide 94.38 boxes per hour to the warehouse.

Boxes added in 40 hours = rate * time = 94.38 * 40 = 3,775.2

Therefore, the inventory at the end of a 40-hour work week would be:

1245 + 3775.2 = 5020.2 boxes

rate = (50000 - 1245) / time

Simplifying this equation, we get:

time = (50000 - 1245) / rate = 511.64 hours (rounded to two decimal places).

Therefore, it will take approximately 511.64 hours to fill the warehouse to its 50,000 box capacity,

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Final answer:

a) The factory is able to provide approximately 94.38 boxes per hour to the warehouse. b) The inventory at the end of a 40-hour work week would be approximately 3775.2 boxes. c) It will take approximately 522.19 hours to fill the warehouse to its 50,000 box capacity.

Explanation:

a) To find the number of boxes per hour the factory is able to provide to the warehouse, we need to calculate the rate of change in the number of boxes over the 8-hour work day. The initial inventory is 1245 boxes, and the final inventory is 2000 boxes. The rate of change can be found by subtracting the initial inventory from the final inventory and dividing by the number of hours: (2000 - 1245) / 8 = 94.38 boxes per hour. Therefore, the factory is able to provide approximately 94.38 boxes per hour to the warehouse.

b) To find the inventory at the end of a 40-hour work week, we need to calculate the rate of change in the number of boxes per hour and then multiply by 40 hours. Using the rate of change calculated in part (a) (94.38 boxes per hour), we can multiply it by 40: 94.38 * 40 = 3775.2 boxes. Therefore, the inventory at the end of a 40-hour work week would be approximately 3775.2 boxes.

c) To find how long it will take to fill the warehouse to its 50,000 box capacity, we can set up a proportion using the rate of change calculated in part (a). Let x represent the number of hours it will take to fill the warehouse to capacity. The proportion is: 94.38 boxes per hour = (50,000 - 1245) boxes / x hours. Solving for x, we can cross-multiply and divide: x = (50,000 - 1245) / 94.38 = 522.19 hours. Therefore, it will take approximately 522.19 hours to fill the warehouse to its 50,000 box capacity.

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