Exercises

Precalculus for Everybody

Chapter 4. Real Functions

Function Applications

  1. (Population growth). The population of a city, \(t\) years after 2000, is:

    \[ P(t)= 60,000\, \mathrm{e}^{0.05t} \text{ inhabitants} \]

    1. Compute the population of the city at the end of the year 2015.

    2. Find the annual percentage of the population growth.

  2. (Depreciation). The value of a machine, \(t\) years after it was bought, is:

    \[ V(t) = A\,\mathrm{e}^{-0.25t} \]

    The machine was bought 9 years ago for \(\$ 150,000\).

    1. How much is it worth today?

    2. Find the annual percentage of the depreciation of his value.

  3. (Population growth). The population of a country in \(t\) years will be:

    \[ P(t) = 18\,\mathrm{e}^{0.02t} \text{ million of inhabitants.} \]
    \[ \begin{aligned} P(t) = 18\,\mathrm{e}^{0.02t} &\text{ million} \\[.5em] &\text{ of inhabitants.} \end{aligned} \]

    1. What is the current population of the country?

    2. Find the population after 15 years.

    3. Find the annual percentage of the population growth.

  4. (Bacterial population growth). An experiment to study a bacterial population growth started with 4,000 bacteria. 10 minutes later, they were 12,000. If the growth is given by the function \(f(t)=A\,\mathrm{e}^{kt}\), find the number of bacteria after 30 minutes?

  5. (Profits growth). The profits of a company increases exponentially according to the function \(f(t)=A\,\mathrm{e}^{kt}\). In 1995 they reached the value of 3 million dollars, and 4.5 million dollars in 2000. Find the profits by the year 2005.

  6. (Radioactive Decay). The quantity \(Q(t)\) of a radioactive substance that remains after \(t\) years is given by:

    \[ Q(t) = A\,\mathrm{e}^{ -0.00015t } \text{ grams} \]

    If there are 3,000 grams left after 5000 years, how many grams were there originally?

  7. (Radioactive decay). A radioactive substance decays according to the function \(f(t)=A\,\mathrm{e}^{-kt}\). Originally, there were 450 grams; 60 years later, there were 400 grams left.

    Find the quantity, in grams, after 240 years.

  8. (National product)}. The national product (N.P.) of a certain country, \(t\) years after 1995, is \(f(t)\) million dollars, where:

    \[ f(t) = P(10)^{kt} \; ,P \, \text{ and } \, k \text{ are constants} \]
    \[ \begin{aligned} f(t) &= P(10)^{kt}, \\[.5em] &\hspace{2em} P \, \text{ and } \, k \, \text{ are constants} \end{aligned} \]

    In 1995 the N.P. was 8,000 million dollars, and it was 16,000 million dollars by the year 2000. Find the N.P. by the year 2010.

  9. (Atmospheric pressure). Atmospheric pressure is \(P(h)\) pounds per square feet at an altitude of \(h\) feet above sea level, where:

    \[ P(h) = M\,\mathrm{e}^{-0.00003h}, \;M \text{ is a constant} \]
    \[ \begin{aligned} P(h) &= M\,\mathrm{e}^{-0.00003h}, \\[.5em] &\hspace{2em} M \text{ is a constant} \end{aligned} \]

    If the atmospheric pressure at sea level is 2,116 pounds per square feet, find the atmospheric pressure outside a plane flying at an altitude of 12,000 feet.

  10. (Light bulb life). A light bulb manufacturer discovers that the share \(f(t)\) of bulbs that still work after \(t\) months of use is given by:

    \[ f(t) = \mathrm{e}^{-0.2t} \]

    1. What percentage of light bulbs last at least 1 month?

    2. What percentage of light bulbs last at least 2 months?

    3. What percentage of light bulbs fail during the second month?

  11. (Product sales). A transnational company knows that if they give away \(x\) thousand of free products to influencers, the sales will be given by:

    \[ V(x) = 30 - 18\, \mathrm{e}^{-0.3x} \text{ thousand of products sold} \]
    \[ \begin{aligned} V(x) = 30 - 18\, &\mathrm{e}^{-0.3x} \text{ thousand} \\[.5em] &\hspace{1em} \text{of products sold} \end{aligned} \]

    1. Find the number of products sold if they don't give any free product to influencers.

    2. Find the number of products sold if they give away 800 free products to influencers.

  12. (Depreciation). The value of a machine after \(t\) years of use is:

    \[ V(t) = 520\, \mathrm{e}^{-0.15t} + 460 \text{ thousand dollars} \]
    \[ \begin{aligned} V(t) = 520\, \mathrm{e}^{-0.15t} &+ 460 \text{ thousand} \\[.5em] &\hspace{1em}\text{ dollars} \end{aligned} \]

    1. Sketch the graph of the function \(V(t)\).

    2. Find the value of the machine when it was brand new.

    3. Find the value of the machine after 20 years of use.

  13. (Radioactive decay). \(Q_0\) is the initial amount of a radioactive substance disintegrating according to \(Q(t)=Q_0\,\mathrm{e}^{-kt}\). The half-life of the substance is \(\lambda\) time units (years, months, hours, etc.). Prove that \(Q(t)\), the amount that remains after \(t\) time units, is:

    \[ Q(t) = Q_0\,\mathrm{e}^{ - \left( \frac{\ln 2}{\lambda} \right) t } \]

  14. (Radioactive decay). The radium is a radioactive element with a half-life of 1,690 years. How long will it take for 200 grams of radium to be reduced to 40 grams?.

    Hint: See the previous problem.

  15. (Blood alcohol level). After drinking whiskey, the level of blood alcohol level of a bus-driver is 0.4 milligrams per milliliter (\(mg/ml\)). The blood alcohol level decreases according to the function:

    \[ f(t) =(0.4) \left( \frac{1}{2} \right)^t, \]

    where \(t\) is the number of hours elapsed since he reached the above level. The legal limit to drive a vehicle is 0.08 \(mg/ml\). How long will the bus-driver have to wait for driving without breaking the law?

  16. (Estimation of the amount). An amount of 12 million dollars is invested at interest rate of \(14\%\) per year, compounding continuously. Find the time needed to obtain 21 million.

  17. (Subscribers). Two Youtube channels compete for subscribers from a specific audience. One of them has 500,000 subscribers and is growing at a monthly rate of \(1.5\%\). The other one has 900,000 subscribers and is decreasing at a monthly rate of \(0.5\%\). How long will it take for both channels to have the same quantity of subscribers?

  18. (Book sales). A new Data Science book will be in the market soon. It is expected that if \(x\) thousands of copies are delivered to the teachers, as presents, the sales in the first year will be \(f(x)=12-5\,\mathrm{e}^{-0.2x}\) thousands of copies. How many copies will have to be given as presents to sale 9,000 copies in the first year?

  19. (National product). The national product (N.P.) of a country grows exponentially. In 1995 it was 60,000 million, and it was 70,000 million by the year 2000. Find the N.P. of 2005.

  20. (World population forecast). The world population was 4,917 million in 1986, and it grows at a rate of 1.65% annually. If the rate keeps constant, how many years will it take for the world population to reach 8,000 millions?

  21. (Age of a fossil). An archaeologist estimated that the amount of \(^{14}\)C in a fossilized tree trunk represents a quarter of the amount of \(^{14}\)C in present-day trees. Find the age of the fossilized tree trunk.

  22. (Estimation of the amount). An investment of \(\$ 7,500,000\) is made at a annual rate of \(28\%\). Find the balance in the account after 2 years, if the interest:

    1. is simple.

    2. is compounding annually.

    3. is compounding quarterly (every 3 months).

    4. is compounding monthly.

    5. is compounded continuously.

  23. (Estimation of the principal). How much money invested will produce \(\$2,500,000\) in 5 years if the rate is \(16\%\) annually, and is compounded:

    1. quarterly?.

    2. continuously?.

  24. (Estimation of the amount). In 1626, the Dutch colonial governor, Peter Minuit, bought the Manhattan Island (New York) from the Native Americans for a mere 24 dollars. Assume that the natives invested those 24 dollars in a bank at a annual rate of 5% compounding continuously. How much money would they have had by the year 2000?

  25. (Time to double an investment). How long will it take for an investment to double in value if the interest rate is 15% per year, compounding:

    1. semiannually?.

    2. continuously?.

  26. (Time to triple an investment). How long will it take for an investment to triple in value if the interest rate is 15% per year, compounding:

    1. semiannually?.

    2. continuously?.

  1.  

    1. \(127,020\)

    2. \(5,127 \%\)

  2.  

    1. \(\$ 15,809.88\)

    2. \(22.12 \%\)

  3.  

    1. \(18\) million

    2. \(24.3\) million

    3. \(2.02 \%\)

  4. \[ 108,000 \]

  5. \(6.75\) million

  6. \[ 6,351 \, gr \]
  7. \[ 280.93 \, gr \]

  8. \(64,000\) million

  9. \[ 1,476.28\; pound/feet^2 \]
  10.  

    1. \(81.87 \%\)

    2. \(67.03\%\)

    3. \(14.84\%\)

  11.  

    1. \(12\) thousand

    2. \(15,841\)

  12.  

    1. \(980\) thousand

    2. \(\$ 485,889.27\)

  1. \(3,924\) years

  2. \(2.32\) horas

  3. \(4\) years

  4. \(29.54\) moths

  5. \(2,554\) books

  6. \(81,666,666\) million

  7. \(29.5\) years

  8. \(11,460\) years

  9.  

    1. \(11,700,000\)

    2. \(12,288,000\)

    3. \(12,886,396\)

    4. \(13,045,843.42\)

    5. \( 13,130,043 \)

  10.  

    1. \(1,140,967.37\)

    2. \(1,123,322.4\)

  11. \[ \$ 3,173,350,575 \]
  12.  

    1. \(4,792\) years

    2. \(4,621\) years

  13.  

    1. \(7,595\) years

    2. \(7,324\) years