# 16.3: Exercises

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## 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.

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?

$$2.29$$ 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.

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?

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 $$2000$$?

$$67.8\%$$ 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?

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?

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 $$\20,000$$?
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
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?
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