# Problem of the Week Problem B and Solution Joey Prepares for Winter

## Problem

Joey the chipmunk will soon be hibernating, so he’s gathering acorns, his food supply for the long winter months.

Joey has four acorns remaining from the previous day, and has gathered acorns over the last few hours as shown in the following table.

 Hour Total Number of Acorns $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$4$$ $$20$$ $$36$$ $$52$$ $$68$$ $$84$$
1. Is the total number of acorns a linear growing pattern? Verify your answer by creating a graph.

2. Suppose Joey continues collecting acorns at this same rate.

1. How many acorns would Joey have collected by the end of Hour $$12$$?

2. How many hours would it take him to collect at least $$330$$ acorns?

3. Write an algebraic expression to represent the total number of acorns Joey would have after collecting for $$n$$ hours.

## Solution

1. Looking at the data, we see that the number of acorns increases by the same amount each hour; Joey is collecting acorns at a rate of 16 per hour. So we expect that the pattern of the total number of acorns is a growing linear pattern. This is verified by the following graph.

1. Hour $$12$$ is $$7$$ more hours after Hour $$5$$. Since Joey will collect $$16$$ acorns in each of those hours, he will have $$7\times 16=112$$ more acorns, giving a total of $$84+112=196$$ acorns by the end of Hour $$12$$.

2. To collect at least $$330$$ acorns in total, Joey needs $$330-196=134$$ more acorns than he has after $$12$$ hours. After $$8$$ more hours, he would have $$8\times 16 = 128$$ more acorns. After $$9$$ more hours, he would have $$9\times 16 = 144$$ more acorns. Therefore, he will need to collect acorns for $$9$$ more hours to get to at least $$330$$ acorns.
Thus, he will need a total of $$12+9=21$$ hours to collect at least $$330$$ acorns.

Alternatively: Joey initially has $$4$$ acorns, so to get to $$330$$ acorns, he needs to collect $$326$$ more acorns. Since he collects $$16$$ acorns per hour, this would take him $$326\div 16= 20\frac{3}{8}$$ hours. This means he will have $$330$$ acorns during the $$21^\textrm{st}$$ hour. That is, he will need to collect for $$21$$ hours to get at least $$330$$ acorns.

3. After $$n$$ hours of collecting $$16$$ acorns each hour, Joey would have $$16\times n$$ acorns. Given that he starts with four leftover acorns, Joey would have a total of $$~(16\times n)+4~$$ acorns.