15.2: Exercises
- Page ID
- 49046
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
-
- \(f(x)=4 \cdot 3^{x}\)
- \(f(x)=5 \cdot 2^{x}\)
- \(f(x)=3200 \cdot 0.1^{x}\)
- \(f(x)=1.5 \cdot 2^{x}\)
- \(f(x)=20 \cdot 5^{x}\)
- \(f(x)=\dfrac{3}{5} \cdot \sqrt{5}^{x}\)
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 \(2014\)
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 \(2061\)
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.
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
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It will take the company \(4.87\) years.
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
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The ant colony has doubled its population after approximately \(23.4\) weeks.
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.
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.