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 Suppose Alice in Wonderland is 185 cm tall (she takes after her father). After eating a piece of mushroom, she starts shrinking, but she shrinks very slowly; with each day that passes, her height is 90% of her height the day before. If we suppose that an ant is 1 mm tall, how many days will have to pass before Alice is shorter than an ant?
Solution: The first thing we have to do is convert Alice's
height from cm to mm. Since 1 cm = 10 mm,
we have 185 cm = 1850 mm.
Next, we let x = number of days. Since we want the ant's height to be greater than Alice's after x days and her shrinking is exponential we have:
1 > (1850)(0.9x)
Now we can take logs of both sides using e as the base:
0 > ln 1850 + (x)(ln 0.9)
0 > 7.5229 + (x)(-0.1056)
-7.5229 > (x)(-0.1056)
71.4015 < x
The inequality switches because we are dividing both sides by a negative number. To err on the side of caution, we round our answer up to 72. So, Alice will be shorter than an ant in 72 days.
 The diamater of circle A is 2 cm, that of circle B, 8 cm. Imagine the diameter of circle A increases at a rate of 10% per minute and that of circle B increases at a rate of 5% per minute. When will the diameter of circle A be greater than that of circle B?
Solution: Let x = number of minutes that have passed. The size of circle A after x minutes is (2)(1.1x), that of circle B, (8)(1.05x), since both are growing exponentially. Since we want circle A to be bigger than circle B after x minutes we have:
1.1x > (4)(1.05x)
by dividing both sides by 2. Taking the logs of both sides using e as the base, we get:
(x)(ln 1.1) > ln 4 + (x)(ln 1.05)
(x)(ln 1.1) - (x)(ln 1.05) > ln 4
(x)(ln 1.1 - ln 1.05) > ln 4
(x)(0.0953 - 0.0488) > 1.3863
(x)(0.0465) > 1.3863
x > 29.8129
To err on the side of caution, we round up our answer. So, circle A will be bigger than circle B after 30 minutes.
 Lower Slobovia is a developing country. Its GDP last year was 75 million zugats. Upper Slobovia is a developed country. Its GDP last year was 300 billion zugats. It enjoys an average annual growth of 3%. If Lower Slobovia wants to have the same annual GDP as Upper Slobovia in 50 years, what does its annual rate of growth have to be?
Solution: Let i = annual rate of growth
of Lower Slobovia. This number is between 0 and 1. The size of
Lower Slobovia's GDP after 50 years is
(75)(1 + i)50. Similarly, the
GDP of Upper Slobovia after 50 years is
happenned to the extra zeroes in the GDP you might ask? Essentially,
what I did was divide both GDPs by
(75)(1 + i)50 =
(1 + i)50 = (4,000)(1.0350)
dividing both sides by 75
(1 + i)50 = 17,535.6241
Now we take the logs of both sides using e as the base.
(50)ln(1 + i) = 9.7720
ln(1 + i) = 0.1954
Now we take the exponent of both sides, again using e as the base:
1 + i = e0.1954
1 + i = 1.2158
i = 0.2158
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