# 16.3: Exercises

- Page ID
- 49054

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.

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.

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.

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.

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 \(2000\).

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

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.

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**-
- \(\$862.90\)
- \(\$1,564.75\)
- \(\$1,566.70\)
- \(\$541.46\)
- \(\$6,242.86\)
- \(\$1,654.22\)
- \(\$910.24\)

- 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**-
- \(P = \$1,484.39\)
- \(P = \$2, 938.67\)
- \(P = \$709.64\)
- \(r = 4.23\%\)
- \(r = 4.31\%\)
- \(t ≈ 1.69\) years
- \(t ≈ 3.81\) years
- \(t ≈ 10.27\) years
- \(t ≈ 13.73\) years