Question 569384: Please help!!!! I will praise you!!!
At 9:30 am tomorrow, we open our doors for the last time. The price of every item in the store will be reduced 10% every hour, on the hour, beginning at 10am until everything goes, The stereo calvin would love to buy is $359 dollars.. Calvin has saved $270 for the stereo. Henry told calvin that at 11:01 am, the price of the stereo will be $287.20. Jacob says $290.79 will be the price. What might he expect the price to be at 11:01 and why? Henry claims that since the prices go down 105 every hour, after 10 hours, the stereo will be free. He tells calvin to arrive at the store at 8pm for a free stereo. Do you agree with Henry's reasoning? Explain why and why not. Sales tax is 6%. Assuming he gets no more money, explain to Calvin when he will be able to afford the stereo and how you figured this out. Be sure your work is shown in an organized format so that it can be easily followed
THANKS SOO MUCH!!!!!!!!!!!!!!!!! :DDDDDDDDDDDDDDDD
Found 2 solutions by IWork4Dessert, solver91311:
Answer by IWork4Dessert(60)   (Show Source):
You can put this solution on YOUR website! Wow, what a mouthful!
Let's break this into more reasonable chunks. First of all, you know that you have a $359 stereo and only $270 to buy it with. At 10 AM, the stereo will be only $323.10(divide $359 by 10 and subtract that number from $359). At 11 AM, the stereo will be $290.79.
The reasoning: To find the price after a 10% discount, you need to find 10% of the number and then subtract it from the number.
At 12 PM, the stereo will be $261.72. At this point, Calvin can afford it, but keep in mind that there's 6% sales tax. To find 6% of $261.72, set up a proportion:
6/100=x/261.72.
Keep in mind that 6% is just 6 out of 100, which will be proportional to an amount out of the price. Cross-multiply to get x alone. Then divide both sides by 100 so that x is equal to 15.7. Add $15.70 to $261.72. At just around $277, the stereo is still too expensive.
At 1 PM, the stereo will be $235.55. Calculate the sales tax again:
6/100=x/235.55.
The sales tax on this one is only $14.33. Add this to $235.55. This gets you $249.88.
THEREFORE......
Calvin can afford the stereo at 1:00 PM. Henry's claim is not true, since the percentage of the stereo will change every time the price changes; just like 10% off of a $100 item is $90, 10% off of a $90 item is $81 and so on. The amount dropped isn't set.
Hope this helps!
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website!
The total amount to be paid after the first 10% reduction (considering the 6% sales tax) is:
The second hour
\ \times\ 0.9\ \times\ 1.06)
or
^2\ \times\ 1.06)
and, in general, after  hours:
(0.9)^t(1.06))
Since Calvin's funds are limited to $270.00, we need to solve
(0.9)^t(1.06)\ <\ 270)
for and round up to the next higher integer.
^t\ =\ \frac{270}{(359)(1.06)}\ \approx\ 0.709)
Take the natural log of both sides
^t\ \approx\ \ln(0.709))
Use
\ =\ n\log_b(x))
to write
\ \approx\ \ln(0.709))
then
}{\ln(0.9)}\ \approx\ 3.25)
Round up
So the total amount will be less than Calvin's $270 right after the fourth hour. 10 is the first hour, 11 is the second hour...
So Calvin needs to come in at 1:00 pm when the total amount will be:
(0.9)^4(1.06)\ \approx\ $249.67)
Which means that Calvin can go to lunch as long as he doesn't spend more than  for his meal including tax and tip.
The only flaw in Calvin's plan is: What if there are a limited number of these stereo sets -- a number less than or equal to the number of people who come in at noon with $280 in their pocket? In such an event, Calvin is out of luck.
And by the way, Henry is an idiot. You take 10% off the CURRENT price, not the FULL price, each hour.
Super Double Plus Extra Credit
Theoretically, the price will never reach zero since after any reduction, 90 percent of the previous amount remains. However, a point will be reached eventually such that the total amount, including tax, will be reduced to an amount when rounded to the nearest one cent will be $0.00. Using the process outlined above, how many hours must elapse before he would actually receive the item absolutely free? Hint: What value of t makes the total price less than $0.005?
John
My calculator said it, I believe it, that settles it
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