[3000] 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)(0.9^{x})

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.

[3001] 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)(1.1^{x}), that of circle B,
(8)(1.05^{x}), since both are
growing exponentially. Since we want circle A to be bigger than circle B
after x minutes we have:

(2)(1.1^{x}) >
(8)(1.05^{x})

or

1.1^{x} >
(4)(1.05^{x})

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.

[3002] 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

We have:

(75)(1 + i)^{50} =
(300,000)(1.03^{50})

(1 + i)^{50} =
(4,000)(1.03^{50})

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 = e^{0.1954}

1 + i = 1.2158

i = 0.2158

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