# 15.2: Exercises

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## Exercise $$\PageIndex{1}$$

Assuming that $$f(x)=c\cdot b^x$$ is an exponential function, find the constants $$c$$ and $$b$$ from the given conditions.

1. $$f(0)=4, \quad f(1)=12$$
2. $$f(0)=5, \quad f(3)=40$$
3. $$f(0)=3,200, \quad f(6)=0.0032$$
4. $$f(3)=12, \quad f(5)=48$$
5. $$f(-1)=4, \quad f(2)=500$$
6. $$f(2)=3, \quad f(4)=15$$
1. $$f(x)=4 \cdot 3^{x}$$
2. $$f(x)=5 \cdot 2^{x}$$
3. $$f(x)=3200 \cdot 0.1^{x}$$
4. $$f(x)=1.5 \cdot 2^{x}$$
5. $$f(x)=20 \cdot 5^{x}$$
6. $$f(x)=\dfrac{3}{5} \cdot \sqrt{5}^{x}$$

## Exercise $$\PageIndex{2}$$

The number of downloads of a certain software application was $$8.4$$ million in the year $$2005$$ and $$13.6$$ million in the year $$2010$$.

1. Assuming an exponential growth for the number of downloads, find the formula for the downloads depending on the year $$t$$.
2. Assuming the number of downloads will follow the formula from part (a), what will the number of downloads be in the year $$2015$$?
3. In which year will the number of downloaded applications reach the $$20$$ million barrier?
1. $$y=8.4 \cdot 1.101^{t}$$ with $$t = 0$$ corresponding to the year $$2005$$
2. approx. $$22.0$$ million
3. It will reach $$20$$ million in the year $$2014$$

## Exercise $$\PageIndex{3}$$

The population size of a city was $$79,000$$ in the year $$1990$$ and $$136,000$$ in the year $$2005$$. Assume that the population size follows an exponential function.

1. Find the formula for the population size.
2. What is the population size in the year $$2010$$?
3. What is the population size in the year $$2015$$?
4. When will the city reach its expected maximum capacity of $$1,000,000$$ residents?
1. $$y=79000 \cdot 1.037^{t}$$ with $$t = 0$$ corresponding to the year $$1990$$
2. approx. $$163, 400$$
3. approx. $$195, 900$$
4. The city will reach maximum capacity in the year $$2061$$

## Exercise $$\PageIndex{4}$$

The population of a city decreases at a rate of $$2.3\%$$ per year. After how many years will the population be at $$90\%$$ of its current size? Round your answer to the nearest tenth.

The city will be at $$90\%$$ of its current size after approximately $$4.5$$ years.

## Exercise $$\PageIndex{5}$$

A big company plans to expand its franchise and, with this, its number of employees. For tax reasons it is most beneficial to expand the number of employees at a rate of $$5\%$$ per year. If the company currently has $$4,730$$ employees, how many years will it take until the company has $$6,000$$ employees? Round your answer to the nearest hundredth.

It will take the company $$4.87$$ years.

## Exercise $$\PageIndex{6}$$

An ant colony has a population size of $$4,000$$ ants and is increasing at a rate of $$3\%$$ per week. How long will it take until the ant population has doubled? Round your answer to the nearest tenth.

The ant colony has doubled its population after approximately $$23.4$$ weeks.

## Exercise $$\PageIndex{7}$$

Add exercises text here.The size of a beehive is decreasing at a rate of $$15\%$$ per month. How long will it take for the beehive to be at half of its current size? Round your answer to the nearest hundredth.

It will take $$4.27$$ months for the beehive to have decreased to half its current size.

## Exercise $$\PageIndex{8}$$

If the population size of the world is increasing at a rate of $$0.5\%$$ per year, how long does it take until the world population doubles in size? Round your answer to the nearest tenth.

It will take $$139.0$$ years until the world population has doubled.