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15.2: Exercises

Last updated

May 2, 2022
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- 15.1: Applications of exponential functions
- 16: Half-life and Compound Interest

Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

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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.

$f(0)=4, \quad f(1)=12$
$f(0)=5, \quad f(3)=40$
$f(0)=3,200, \quad f(6)=0.0032$
$f(3)=12, \quad f(5)=48$
$f(-1)=4, \quad f(2)=500$
$f(2)=3, \quad f(4)=15$

Answer

$f(x)=4 \cdot 3^{x}$
$f(x)=5 \cdot 2^{x}$
$f(x)=3200 \cdot 0.1^{x}$
$f(x)=1.5 \cdot 2^{x}$
$f(x)=20 \cdot 5^{x}$
$f(x)=\dfrac{3}{5} \cdot \sqrt{5}^{x}$

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 $2010$ .

Assuming an exponential growth for the number of downloads, find the formula for the downloads depending on the year $t$ .
Assuming the number of downloads will follow the formula from part (a), what will the number of downloads be in the year $2015$ ?
In which year will the number of downloaded applications reach the $20$ million barrier?

Answer

$y=8.4 \cdot 1.101^{t}$ with $t = 0$ corresponding to the year $2005$
approx. $22.0$ million
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.

Find the formula for the population size.
What is the population size in the year $2010$ ?
What is the population size in the year $2015$ ?
When will the city reach its expected maximum capacity of $1,000,000$ residents?

Answer

$y=79000 \cdot 1.037^{t}$ with $t = 0$ corresponding to the year $1990$
approx. $163, 400$
approx. $195, 900$
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.

Answer: 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.

Answer: 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.

Answer: 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.

Answer: 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.

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

Exercise 15.2.1\PageIndex{1}

Exercise 15.2.2\PageIndex{2}

Exercise 15.2.3\PageIndex{3}

Exercise 15.2.4\PageIndex{4}

Exercise 15.2.5\PageIndex{5}

Exercise 15.2.6\PageIndex{6}

Exercise 15.2.7\PageIndex{7}

Exercise 15.2.8\PageIndex{8}

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Exercise $\PageIndex{1}$

Exercise $\PageIndex{2}$

Exercise $\PageIndex{3}$

Exercise $\PageIndex{4}$

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Exercise $\PageIndex{6}$

Exercise $\PageIndex{7}$

Exercise $\PageIndex{8}$