16.3: Exercises
- Page ID
- 49054
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)An unstable element decays at a rate of \(5.9\%\) per minute. If \(40\)mg of this element has been produced, then how long will it take until \(2\)mg of the element are left? Round your answer to the nearest thousandth.
- Answer
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It takes \(49.262\) minutes until \(2\) mg are left of the element.
A substance decays radioactively with a half-life of \(232.5\) days. How much of \(6.8\) grams of this substance is left after \(1\) year?
- Answer
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\(2.29\) grams are left after \(1\) year.
Fermium-252 decays in \(10\) minutes to \(76.1\%\) of its original mass. Find the half-life of fermium-252.
- Answer
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The half-life of fermium-252 is \(25.38\) minutes.
How long do you have to wait until \(15\)mg of beryllium-7 have decayed to \(4\)mg, if the half-life of beryllium-7 is \(53.12\) days?
- Answer
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You have to wait approximately \(101.3\) days.
If Pharaoh Ramses II died in the year \(1213\) BC, then what percent of the carbon-14 was left in the mummy of Ramses II in the year \(2000\)?
- Answer
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\(67.8\%\) of the carbon-14 is left in the year \(2000\).
In order to determine the age of a piece of wood, the amount of carbon-14 was measured. It was determined that the wood had lost \(33.1\%\) of its carbon-14. How old is this piece of wood?
- Answer
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The wood is approximately \(3323\) years old
Archaeologists uncovered a bone at an ancient resting ground. It was determined that \(62\%\) of the carbon-14 was left in the bone. How old is the bone?
- Answer
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The bone is approximately \(3952\) years old.
An investment of \(\$5,000\) was locked in for \(30\) years. According to the agreed conditions, the investment will be worth \(\$5,000\cdot 1.08^{t}\) after \(t\) years.
- How much is the investment worth after \(5\) years?
- After how many years will the investment be worth \(\$20,000\)?
- Answer
-
- \(\$7, 346.64\)
- It takes approximately \(18\) years
Determine the final amount in a savings account under the given conditions.
- \(\$700\),& compounded quarterly, & at \(3\%\), & for \(7\) years
- \(\$1400\),& compounded annually, & at \(2.25\%\), & for \(5\) years
- \(\$1400\),& compounded continuously, & at \(2.25\%\), & for \(5\) years
- \(\$500\),& compounded monthly, & at \(3.99\%\), & for \(2\) years
- \(\$5000\),& compounded continuously, & at \(7.4\%\), & for \(3\) years
- \(\$1600\),& compounded daily, & at \(3.333\%\), & for \(1\) year
- \(\$750\),& compounded semi-annually, & at \(4.9\%\), & for \(4\) years
- Answer
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- \(\$862.90\)
- \(\$1,564.75\)
- \(\$1,566.70\)
- \(\$541.46\)
- \(\$6,242.86\)
- \(\$1,654.22\)
- \(\$910.24\)
- Find the amount \(P\) that needs to be invested at a rate of \(5 \%\) compounded quarterly for \(6\) years to give a final amount of \(\$ 2000\).
- Find the present value \(P\) of a future amount of \(A=\$ 3500\) invested at \(6 \%\) compounded annually for \(3\) years.
- Find the present value \(P\) of a future amount of \(\$ 1000\) invested at a rate of \(4.9 \%\) compounded continuously for \(7\) years.
- At what rate do we have to invest \(\$1900\) for \(4\) years compounded monthly to obtain a final amount of \(\$2250\)?
- At what rate do we have to invest \(\$1300\) for \(10\) years compounded continuously to obtain a final amount of \(\$2000\)?
- For how long do we have to invest \(\$3400\) at a rate of \(5.125 \%\) compounded annually to obtain a final amount of \(\$3700\)?
- For how long do we have to invest \(\$1000\) at a rate of \(2.5 \%\) compounded continuously to obtain a final amount of \(\$1100\)?
- How long do you have to invest a principal at a rate of \(6.75\%\) compounded daily until the investment doubles its value?
- An certain amount of money has tripled its value while being in a savings account that has an interest rate of \(8\%\) compounded continuously. For how long was the money in the savings account?
- Answer
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- \(P = \$1,484.39\)
- \(P = \$2, 938.67\)
- \(P = \$709.64\)
- \(r = 4.23\%\)
- \(r = 4.31\%\)
- \(t ≈ 1.69\) years
- \(t ≈ 3.81\) years
- \(t ≈ 10.27\) years
- \(t ≈ 13.73\) years