16.3: Exercises

Exercise \(\PageIndex{1}\)

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

It takes \(49.262\) minutes until \(2\) mg are left of the element.

Exercise \(\PageIndex{2}\)

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

\(2.29\) grams are left after \(1\) year.

Exercise \(\PageIndex{3}\)

Fermium-252 decays in \(10\) minutes to \(76.1\%\) of its original mass. Find the half-life of fermium-252.

Answer

The half-life of fermium-252 is \(25.38\) minutes.

Exercise \(\PageIndex{4}\)

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

You have to wait approximately \(101.3\) days.

Exercise \(\PageIndex{5}\)

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

\(67.8\%\) of the carbon-14 is left in the year \(2000\).

Exercise \(\PageIndex{6}\)

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

The wood is approximately \(3323\) years old

Exercise \(\PageIndex{7}\)

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

The bone is approximately \(3952\) years old.

Exercise \(\PageIndex{8}\)

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.

1. How much is the investment worth after \(5\) years?
2. After how many years will the investment be worth \(\$20,000\)?

Answer

1. \(\$7, 346.64\)
2. It takes approximately \(18\) years

Exercise \(\PageIndex{9}\)

Determine the final amount in a savings account under the given conditions.

1. \(\$700\),& compounded quarterly, & at \(3\%\), & for \(7\) years
2. \(\$1400\),& compounded annually, & at \(2.25\%\), & for \(5\) years
3. \(\$1400\),& compounded continuously, & at \(2.25\%\), & for \(5\) years
4. \(\$500\),& compounded monthly, & at \(3.99\%\), & for \(2\) years
5. \(\$5000\),& compounded continuously, & at \(7.4\%\), & for \(3\) years
6. \(\$1600\),& compounded daily, & at \(3.333\%\), & for \(1\) year
7. \(\$750\),& compounded semi-annually, & at \(4.9\%\), & for \(4\) years

Answer

1. \(\$862.90\)
2. \(\$1,564.75\)
3. \(\$1,566.70\)
4. \(\$541.46\)
5. \(\$6,242.86\)
6. \(\$1,654.22\)
7. \(\$910.24\)

Exercise \(\PageIndex{10}\)

1. 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 \(\$ 2000\).
2. Find the present value \(P\) of a future amount of \(A=\$ 3500\) invested at \(6 \%\) compounded annually for \(3\) years.
3. Find the present value \(P\) of a future amount of \(\$ 1000\) invested at a rate of \(4.9 \%\) compounded continuously for \(7\) years.
4. At what rate do we have to invest \(\$1900\) for \(4\) years compounded monthly to obtain a final amount of \(\$2250\)?
5. At what rate do we have to invest \(\$1300\) for \(10\) years compounded continuously to obtain a final amount of \(\$2000\)?
6. For how long do we have to invest \(\$3400\) at a rate of \(5.125 \%\) compounded annually to obtain a final amount of \(\$3700\)?
7. For how long do we have to invest \(\$1000\) at a rate of \(2.5 \%\) compounded continuously to obtain a final amount of \(\$1100\)?
8. How long do you have to invest a principal at a rate of \(6.75\%\) compounded daily until the investment doubles its value?
9. 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

1. \(P = \$1,484.39\)
2. \(P = \$2, 938.67\)
3. \(P = \$709.64\)
4. \(r = 4.23\%\)
5. \(r = 4.31\%\)
6. \(t ≈ 1.69\) years
7. \(t ≈ 3.81\) years
8. \(t ≈ 10.27\) years
9. \(t ≈ 13.73\) years