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.
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 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:
(2)(1.1x) > (8)(1.05x)
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
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.
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 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 (300,000)(1.03)50. What happenned to the extra zeroes in the
GDP you might ask? Essentially, what I did was divide both GDPs by 1,000,000 to make the computations a little easier. Who needs extra zeroes when you don't need them?
(75)(1 + i)50 = (300,000)(1.0350)
(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
So, for Lower Slobovia to have the same GDP as Upper Slobovia in 50 years, it must have an annual growth rate of 21.58%
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