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15.2: Exercises

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    49046

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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. \(f(2)=3, \quad f(4)=15\)
    Answer
    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?
    Answer
    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?
    Answer
    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.

    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.

    This page titled 15.2: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.