A giant cookie sits on a plate. Cookie Monster eats half the cookie with one bite. With another bite he eats half of the remaining cookie.
Cookie Monster keeps taking bites, eating half the remaining cookie with each bite.
(a) After 5 bites, what fraction of the cookie has Cookie Monster eaten?
(b) How many bites does it take before less than one one-thousandth of the cookie remains?
After one bite, half the cookie remains. After two bites, one-fourth of the cookie remains. With each bite, the remaining portion of cookie gets halved.
Let's put this information in a table:
After the nth bite, the fraction of cookie remaining is .
(a) After 5 bites, the fraction of cookie remaining is
This means Cookie Monster has eaten of the cookie.
(b) We want to find the first value of n for which
Since , we need to solve the inequality
Multiplying both sides by 2n and by 1000, we have
1000 < 2n
log 2 1000 < n
9.966 < n
In order for n to be a whole number greater than 9.966, we must have n = 10. To make sure, we can check the values of a9 and a10.
It takes 10 bites before less than one one-thousandth of the cookie remains.
A dress is listed at $150. At the end of each week the price is reduced by 10%.
(a) What is the price of the dress after 8 weeks?(b) After how many weeks will the dress be less than $50?
(hint: If 10% of the price is taken off, then 90% of the price remains.)
After one week, the price will be reduced by 10%. This means the dress will cost 90% of its original price, so it will cost
0.9(150) = 135.
After two weeks the price will be reduced by 10% again, so the dress will cost
0.9(135) = (0.9)(0.9)(150).
Continuing in this fashion, after n weeks the price of the dress will be
an = (.9)n(150).
(a) After 8 weeks, the price of the dress will be
a8 = (0.9)8(150) = 64.57.
The dress will cost $64.57.
(b) This question is asking for what value of n we have an<50.
At this point it's tempting to take logs of both sides.
If we do that, we get
n < 10.4.
That doesn't make sense, because we're looking for a statement of the form
n > ...
The weirdness is because is negative, and negative signs do funny things to inequalities. To avoid the negatives, let's rearrange the fractions before taking the log.
Since n must be a whole number, we can conclude that after n = 11 weeks the dress will be less than $50.
A dress is listed at $75. At the end of each week the price is reduced by $2.
This problem describes an arithmetic sequence. The dress initially costs a1 = 75. Each week the price is reduced by $2, so d = -2. After n weeks the price of the dress will be
an = 75 + (n – 1)(-2).
(a) After 8 weeks the price of the dress is
a8 = 75 + (7)(-2) = 61.
The dress will be $61.
(b) We want to find the first n for which an<50.
75 + (n – 1)(-2) < 50
75 – 2n + 2 < 50
27 < 2n
13.5 < n
After 14 weeks the dress will be less than $50. We could confirm this answer by calculating a13 = 51 and a14 = 49.
During her first week of biking, Jenny bikes 3km per day. Each week, she bikes 3km farther per day than she did the previous week.
(a) How many weeks does it take Jenny to get up to 15km per day?(b) How far does Jenny bike on her 31st day?
This problem describes an arithmetic sequence. During her 1st week Jenny bikes a1 = 3 km per day. Each week she adds d = 3 km to her distance. During her nth week she bikes
an = 3 + (n – 1)(3)km per day.
(a) This question is asking for the smallest n with an = 15.
15 = 3 + (n – 1)3
12 = (n – 1)3
4 = n – 1
5 = n
During her 5th week of biking, Jenny will go 15 km per day.
(b) This is a slightly sneaky question, since it's phrased in terms of days instead of weeks. Her 31st day will be during her 5th week of biking, so Jenny will go 15 km that day.
A particular drug decays in the system so that 1 day after taking a dose, 30% of the drug remains in the bloodstream. Paco takes one pill.
(a) What percent of the drug remains in Paco's bloodstream 5 days after taking one dose?(b) How many days before less than 1% of the drug remains in Paco's bloodstream?
This problem describes a geometric sequence. One day after taking a dose, 30% of the drug remains. Two days after, 30% of that 30%, or (.3)2, remains. After n days, (.3)n of the drug remains.
(a) After 5 days, the portion of the drug in Paco's bloodstream is
a5 = (.3)5 = .00243
or 0.2 %.
(b) The question is asking for the smallest n with an<.01. As with question 2(b), we rearrange numbers before taking logs.
After 4 days, less than 1% of the drug will remain in Paco's bloodstream.
Make it rain.