SOLUTION: Please help me fill in the blanks: Andrew and Bobby both opened restaurants at the same time. Andrew reported that on the first week after opening his restaurant, he had a profit

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Question 1043740: Please help me fill in the blanks:
Andrew and Bobby both opened restaurants at the same time. Andrew reported that on the first week after opening his restaurant, he had a profit of $500, and every week after that, his profit continually increased by 12%. Bobby reported that on the first week after opening his restaurant, he had a profit of $600, and every week after that, his profit continually increased by 8%.
Use the information given above to determine how long Bobby will have a higher profit than Andrew.
The first week Andrew will have a higher profit than Bobby will occur ___ weeks after he opens his restaurant.
The first week when Andrew's profit is higher than Bobby's, Bobby starts an advertising campaign that makes his weekly profit begin to increase by 15%. Taking this change into account, what will Bobby's weekly profit be 14 weeks after opening his restaurant? ___
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +P+ = profit after +n+ weeks
Let +A+ = the amount of the 1st weeks profit
Let +r+ = weekly increase in profit
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The equation for both of them is:
+P+=+A%2A%28+1+%2B+r+%29%5En+
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Andrew's equation:
+P+=+500%2A%28+1+%2B+.12+%29%5En+
Bobby's equation:
+P+=+600%2A%28+1+%2B+.08+%29%5En+
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Note that the 1st week for each will be +n+=+0+
Find the week when their profits are equal
+500%2A%28+1+%2B+.12+%29%5En+=+600%2A%28+1+%2B+.08+%29%5En+
+500%2A1.12%5En+=+600%2A1.08%5En+
+1.12%5En+%2F+1.08%5En+=+600%2F500+
+%28+1.12%2F1.08+%29%5En+=+1.2+
Take the log of both sides

+n%2A%28+.049218+-+.033424+%29+=+.079181+
+.015794n+=+.079181+
+n+=+5.01336+
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If I start with week 0, this is after +5+%2B+1+=+6+ weeks, they
make the same profit, so
The first week Andrew will have a higher profit than Bobby will occur 7 weeks after he opens his restaurant.
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After 7 weeks, Bobby's profit is:
+P+=+600%2A%28+1+%2B+.08+%29%5E6+ ( 1st week is week 0 )
+P+=+600%2A1.08%5E6+
+P+=+600%2A+1.58687+
+P+=+952.12+
Then Bobby's equation becomes:
+P+=+952.12%2A%28+1+%2B+.15+%29%5En+
+n+ is now from +n+=+6+ to +n+=+13+
This is 7 weeks, but the 1st week is week +0+, so
+P+=+952.12%2A%28+1+%2B+.15+%29%5E6+
+P+=+952.12%2A1.15%5E6+
+P+=+952.12%2A2.31306+
+P+=+2202.31
What is Bobby's weekly profit 14 weeks after opening his restaurant?
$2,202.31
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I could easily get this wrong.
Get another opinion, too.