16.3: Exercises

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Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

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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.

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.

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

Exercise $\PageIndex{9}$

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

$\$862.90$
$\$1,564.75$
$\$1,566.70$
$\$541.46$
$\$6,242.86$
$\$1,654.22$
$\$910.24$

Exercise $\PageIndex{10}$

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

Exercise \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\)

Exercise \(\PageIndex{3}\)

Exercise \(\PageIndex{4}\)

Exercise \(\PageIndex{5}\)

Exercise \(\PageIndex{6}\)

Exercise \(\PageIndex{7}\)

Exercise \(\PageIndex{8}\)

Exercise \(\PageIndex{9}\)

Exercise \(\PageIndex{10}\)