Now You Try

<p>Carbon-14 is used by scientists to estimate how long ago a plant or animal lived. The half-life of carbon-14 is 5730 years. A particular plant contained 100 g of carbon-14 at the time that it died.</p><p>a) How much carbon-14 would remain after 5730 years?</p><p>b) Write an equation to represent the amount of carbon-14 that remains after t years.</p><p>c) After how many years would 80 g of carbon -14 remain?</p><p>d) Estimate the instantaneous rate of change at 10 years.</p>

<p>A 10-mg sample of bismuth-214 decays to 9 mg in 3 min.</p><p>Describe how the graph would change if the half-life were
* i) shorter
* ii) longer
Give reasons for your answers.</p>

<p>Most bacteria reproduce by dividing into identical cells. This process is called binary fission. A certain type of bacteria can double its numbers every 20 minutes. Suppose 100 of these cells are in one culture dish and 250 of the cells are in another culture dish. Write and evaluate an expression that shows the total number of bacteria cells in both dishes after 20 minutes.</p>

<p>A population of yeast cells can double in as little as 1 h. Assume an initial population of 80 cells.</p><p> Use your equation to determine the population after 90 min.</p>

<p>For the given function find</p><p>a) whether it's Exponential Growth or Decay</p><p>b) its initial value</p><p>c) Growth or Decay Rate</p><p><code class='latex inline'>\displaystyle
P(n) = (0.8)^n
</code></p>

<p>The number of bees in a hive is 1000 on June 1 and doubles every month. This can be expressed as <code class='latex inline'>N=1000 \times 2^t</code>, where N represents the number of bees and t represents time, in months.</p>
<ul>
<li>When were there 125 bees? Explain.</li>
</ul>

<p>The intensity of light energy under water decreases rapidly. Many factors affect how quickly the intensity decreases. The light energy under water can be calculated using an exponential relation. For example.</p>
<ul>
<li>Ocean: <code class='latex inline'>I= 325 \times (1.024)^{-d}</code></li>
<li>Lake Erie: <code class='latex inline'>I= 401 \times (1.222)^{-d}</code></li>
</ul>
<p>In these relations, <code class='latex inline'>d</code> is the depth in metres and <code class='latex inline'>I</code> is the light energy, in langleys.</p>
<ul>
<li>In which body of water does the intensity decrease more quickly? Explain why.</li>
</ul>

<p>A computer loses its value each month after it is purchased. Its value as a function of time, in months, is modelled by <code class='latex inline'>V(m) = 1500(0.95)^m</code>.</p>
<ul>
<li>In which month after it is purchased does the computer’s worth fall
below $900?</li>
</ul>

<p>The number of bees in a hive is 1000 on June 1 and doubles every month. This can be expressed as <code class='latex inline'>N=1000 \times 2^t</code>, where N represents the number of bees and t represents time, in months.</p>
<ul>
<li>Is it possible for <code class='latex inline'>t</code> to be -1? What does this mean?</li>
</ul>

<p>Light intensity in a lake falls by 9% per metre of depth relative to the surface. </p><p> Write an equation that models the intensity of light per metre of depth. Assume that the intensity is 100% at the surface.</p>

<p>A 10-mg sample of bismuth-214 decays to 9 mg in 3 min.</p><p> Describe how the graph would change if the initial sample size were</p>
<ul>
<li>i) greater</li>
<li>ii) less</li>
</ul>
<p>Give reasons for your answers.</p>

<p>A student records the internal temperature of a hot sandwich that has been left to cool on a kitchen counter. The room temperature is 19 <code class='latex inline'>^oC</code>. An equation that models this situation is</p><p><code class='latex inline'>\displaystyle
T(t) = 63(0.5)^{\frac{t}{10}} +19
</code></p><p>where Tis the temperature in degrees Celsius and t is the time in minutes.</p><p>How much time did it take for the sandwich to reach an internal
temperature of 30 <code class='latex inline'>^oC</code>?</p>

<p>Use the decay equation for polonium-218, <code class='latex inline'>\displaystyle A(t) = A_0(\frac{1}{2})^{\frac{t}{3.1}}</code>, where <code class='latex inline'>A</code> is the amount remaining after t minutes and <code class='latex inline'>A_0</code> is the initial amount. </p>
<ul>
<li>Would your answer to part b) change if the size of the initial sample were changed? Explain why or why not.</li>
</ul>

<p>Solve the exponential equations.</p><p><code class='latex inline'>\displaystyle
P = 9000(\frac{1}{2})^8
</code></p>

<p>The doubling time for a certain type of yeast cell is 3 h. The number of cells after <code class='latex inline'>t</code> hours is described by <code class='latex inline'>N(t) = N_{0}2^{\frac{t}{3}}</code>, where <code class='latex inline'>N_{0}</code> is the initial population.</p><p>(a) How would the graph and the equation change if the doubling time were 9 h?</p><p>(b) What are the domain and range of this function in the context of this problem?</p>

<p>A computer loses its value each month after it is purchased. Its value as a function of time, in months, is modelled by <code class='latex inline'>V(m) = 1500(0.95)^m</code>.</p>
<ul>
<li>What is the rate of depreciation? Explain how you know.</li>
</ul>

<p>Chris walks halfway along a 100-m track in <code class='latex inline'>1</code> min. then half of the remaining distance in the next minute, then half of the remaining distance in the third minute, and so on.</p>
<ul>
<li>How far has Chris walked after 10 min?</li>
</ul>

<p>The half-life of a certain substance is 3.6 days. How long will it take for 20 g of the substance to decay to 7 g?</p>

<p>Alchemy is the ancient study of turning substances into gold using chemical, or sometimes magical, means. When platinum-197 undergoes a nuclear process known as beta decay, it becomes gold-197, which is a stable isotope. The half-life of platinum-197 is 20 h.</p><p> How long would it take to turn 90% of a l-kg sample of platinum-197 into gold?</p>

<p>A computer loses its value each month after it is purchased. Its value as a function of time, in months, is modelled by <code class='latex inline'>V(m) = 1500(0.95)^m</code>.</p>
<ul>
<li>What is the initial value of the computer? Explain how you know.</li>
</ul>

<p>Inflation The formula <code class='latex inline'>\displaystyle C=c(1+r)^{n} </code> can be used to estimate the future cost <code class='latex inline'>\displaystyle C </code> of an item due to inflation. Here <code class='latex inline'>\displaystyle c </code> represents the current cost of the item, <code class='latex inline'>\displaystyle r </code> is the rate of inflation, and <code class='latex inline'>\displaystyle n </code> is the number of years for the projection. Suppose a video game system costs <code class='latex inline'>\displaystyle \$ 299 </code> now. How much will the price increase in nine months with an annual inflation rate of <code class='latex inline'>\displaystyle 3.2 \% ? </code></p>

<p>A population of yeast cells can double in as little as l h. Assume an initial population of 80 cells.</p><p> What are the domain and range for this situation?</p>

<p>A town with a population of 12 000 has been growing at an average rate of 2.5% for the last 10 years. Suppose this growth rate will be maintained in the future. The function that models the town’s growth is</p><p><code class='latex inline'>\displaystyle
P(n) = 12(1.025^n)
</code></p><p>where P(n) represents the population (in thousands) and n is the number of years from now.</p><p>a) Determine the population of the town in 10 years.</p><p>b) Determine the number of years until the population doubles.</p><p>c) Use this equation (or another method) to determine the
number of years ago that the population was 8000. Answer to the nearest year.</p><p>d) What are the domain and range of the function?</p>

<p>A population of yeast cells can double in as little as l h. Assume an initial population of 80 cells.</p>
<ul>
<li>Use your equation to determine the population after 6 h.</li>
</ul>

<p>The number of cells in a cell culture grows exponentially. The number of cells in the culture as a function of time is given by the expression <code class='latex inline'>\displaystyle N\left(\frac{6}{5}\right)^{t} </code>, where <code class='latex inline'>\displaystyle t </code> is measured in hours and <code class='latex inline'>\displaystyle N </code> is the initial size of the culture.</p><p>a. After 2 hours, there were 144 cells in the culture. What was <code class='latex inline'>\displaystyle N ? </code></p><p>b. How many cells were in the culture after 20 minutes?</p><p>c. How many cells were in the culture after <code class='latex inline'>\displaystyle 2.5 </code> hours?</p>

<p>A particular radioactive substance has a half-life of 3 years. Suppose an initial sample has a mass of 200 mg. </p><p><strong>NOTE:</strong> To solve for x in exponential equation of <code class='latex inline'>a^x = b</code>, use <code class='latex inline'>x = \frac{\log b}{\log a}</code>.</p><p>a) Write the equation that relates the mass of radioactive material remaining to time.</p><p>b) How much will remain after one decade?</p>

<p>A savings bond offers interest at a rate of <code class='latex inline'>6.6\%</code>, compounded semi-annually. Suppose that a <code class='latex inline'>\$500</code> bond is purchased.</p><p><strong>a)</strong> Write an equation for the value of the investment as a function of time, in years.</p><p><strong>b)</strong> Determine the value of the investment after 5 years.</p><p><strong>c)</strong> How long will it take for the investment to double in value?</p>

<p>A Galápagos cactus finch population increases
by half every decade. The number of finches is modeled by the
expression <code class='latex inline'> 45 \cdot 1.5^{d} </code> , where <code class='latex inline'> d </code> is the number of decades after the
population was measured. Evaluate the expression for <code class='latex inline'> d=-2, d=0 </code> ,
and <code class='latex inline'> d=1 </code> . What does each value of the expression represent in the
situation?</p>

<p>In each case, write an equation that models the situation described. Explain what each part of each equation represents.</p>
<ul>
<li>the population of a colony if a single bacterium takes 1 day to divide into two; the population is P after t days</li>
</ul>

<p>A computer loses its value each month after it is purchased. Its value as a function of time, in months, is modelled by <code class='latex inline'>V(m) = 1500(0.95)^m</code>.</p>
<ul>
<li>Determine the value of the computer after 2 years.</li>
</ul>

<p>Which of these functions describe exponential decay? Explain.</p><p>a) <code class='latex inline'>\displaystyle
g(x) = -4(3)^x
</code></p><p>b) <code class='latex inline'>\displaystyle
h(x) = 0.8(1.2)^x
</code></p><p>c) <code class='latex inline'>\displaystyle
j(x) = 3(0.8)^{2x}
</code></p><p>d) <code class='latex inline'>\displaystyle
j(x) = \frac{1}{3}(0.9)^{\frac{x}{2}}
</code></p>

<p>Half-life is the time it takes for half of a sample of a radioactive element to decay. The function <code class='latex inline'>\displaystyle{M(t)=P\left(\frac{1}{2}\right)^{\frac{t}{h}}}</code> can be used to calculate the mass remaining if the half-life is <code class='latex inline'>h</code> and the initial mass is <code class='latex inline'>P</code>. The half-life of radium is 1620 years.</p><p>b) How many years will it take until the laboratory has only 4 g of radium?</p>

<p>Light intensity in a lake falls by 9% per metre of depth relative to the surface. </p><p> Determine the intensity of light at a depth of 7.5 m.</p>

<p>A flywheel is rotating under friction. The number, <code class='latex inline'>R</code>, of revolutions per minute after t minutes can be determined using the function <code class='latex inline'>R(t) = 4000(0.75)^{2t}</code>.</p><p>Determine the number of revolutions per minute after </p>
<ul>
<li>i) 1 min</li>
<li>ii) 3 min</li>
</ul>

<p>For the given function find</p><p>a) whether it's Exponential Growth or Decay</p><p>b) its initial value</p><p>c) Growth or Decay Rate</p><p><code class='latex inline'>\displaystyle
A(x) = 0.5(3)^x
</code></p>

<p>Suppose that a shelf can hold cylindrical drums with a fixed height of <code class='latex inline'>1m</code>.</p><p>Find the surface area and diameter of a drum with a volume of <code class='latex inline'>0.8 m^3</code>.</p>

<p>A 50-mg sample of cobalt-60 decays to 40 mg after 1.6 min.</p><p><strong>a)</strong> Determine the half-life of cobalt-60.</p><p><strong>b)</strong> How long will it take for the sample to decay to <code class='latex inline'>5\%</code> of its initial amount?</p>

<p>Suppose a square-based pyramids a fixed height of 25 m.</p><p>How should you limit the domain of this function so that the mathematical model fits this situation?</p>

<p>When a patient takes a certain drug, <code class='latex inline'>\frac{1}{10}</code> of the drug that remains in his or her system is used per hour. A patient is given a 500 mg dose of a drug.</p><p>(b) After how many hours will less than 1% of the original mass remain?</p>

<p>A species of bacteria has a population of 500 at noon. It doubles every 10 h, The function that models the growth of the population, P, at any hour, t, is</p><p><code class='latex inline'>\displaystyle
P(t) = 500 (2^{\frac{t}{10}})
</code></p><p>a) why is the exponent <code class='latex inline'>\displaystyle
\frac{t}{10}
</code></p><p>b) Why is the multiplier 500?</p><p>c) Determine the population at midnight.</p><p>d) Determine the population at noon the next day.</p><p>e) Determine the time at which the population first exceeds 2000.</p>

<p>For the given function find</p><p>a) whether it's Exponential Growth or Decay</p><p>b) its initial value</p><p>c) Growth or Decay Rate</p><p><code class='latex inline'>\displaystyle
Q(w) =600(\frac{5}{8})^w
</code></p>

<p>Suppose that a shelf can hold cylindrical drums with a fixed height of 1m.</p><p>Write a simplified equation, using rational exponents where appropriate, to express the surface area in terms of the volume for drums that will fit on the shelf.</p>

<p>The probability, <code class='latex inline'>P</code>, in percent, that a person will respond to a certain advertisement can be modelled by the equation <code class='latex inline'>P(n) = 1 - 2^{-0.05n}</code>, where <code class='latex inline'>n</code> is the number of days since the advertisement began on television. After how many days is the probability greater than 20%?</p>

<p>Chris walks halfway along a 100-m track in 1 min. then half of the remaining distance in the next minute, then half of the remaining distance in the third minute, and so on.</p><p>(c) Write an equation to model this situation.</p>

<p>Suppose a square-based pyramids a fixed height of <code class='latex inline'>25 m</code>.</p><p> Write an equation, using rational exponents where appropriate, to express the side length of the base of a square-based pyramid in terms of its volume.</p>

<p> The growth in population of a small town since 1996 is given by the function <code class='latex inline'>P(n) = 1250(1.03)^n</code>. </p><p>a) What is the initial population? Explain how you know. </p><p>b) What is the growth rate? Explain how you know. </p><p>c) Determine the population in the year 2007. </p><p>d) In which year does the population reach 2000 people?</p>

<p>In 1990, a sum of $1000 is invested at a rate of 6% per year for 15 years.</p><p>What is the initial amount?</p>

<p>A student records the internal temperature of a hot sandwich that has been left to cool on a kitchen counter. The room temperature is 19 <code class='latex inline'>^oC</code>. An equation that models this situation is</p><p><code class='latex inline'>\displaystyle
T(t) = 63(0.5)^{\frac{t}{10}} +19
</code></p><p>where <code class='latex inline'>T</code> is the temperature in degrees Celsius and <code class='latex inline'>t</code> is the time in minutes.</p><p> Determine the temperature, to the nearest degree, of the sandwich after 20 min.</p>

<p>A 10-mg sample of bismuth-214 decays to 9 mg in 3 min.</p>
<ul>
<li>Determine the half-life of bismuth-214.</li>
</ul>

<p>In each case, write an equation that models the situation described. Explain what each part of each equation represents.</p><p> the percent of colour left if blue jeans lose 1% of their colour every time they are washed.</p>

<p>The volume and surface area of a cylinder are given, respectively, by the formulas</p><p><code class='latex inline'>V = \pi r^2 h</code> and <code class='latex inline'>SA = 2\pi rh + 2\pi r^2</code>.</p><p>Calculate the ratio for a radius of 0.8 cm and height of 12 cm.</p>

<p>A town has a population of 8400 in 1990. Fifteen years later, its population grew to 12 500. Determine the average annual growth rate of this town’s population.</p>

<p>Solve each exponential equation. Express answers to the nearest hundredth of a unit.</p><p><code class='latex inline'>A = 250(1.05)^{10}</code></p>

<p>In 1990, a sum of $1000 is invested at a rate of 6% per year for 15 years.</p><p>What is the growth rate?</p>

<p><code class='latex inline'>\displaystyle P_1=1006(1.016)^t</code></p><p><code class='latex inline'>\displaystyle P_2=1000\times2^{\frac{t}{43.5}}</code></p><p><strong>a)</strong> Use both models to predict</p>
<ul>
<li>i) the town’s population after 100 years</li>
<li>ii) how long it will take for the town’s population to reach 20 000</li>
</ul>
<p><strong>b)</strong> Do these models generate predictions that are identical, quite close, or completely different? How would you account for any discrepancies?</p>

<p>A collector’s hockey card is purchased in 1990 for $5. The value increases by 6% every year.</p><p>a) Write an equation that models the value of the card, given the number of years since 1990.</p><p>b) Determine the increase in value of the card in the 4th year after it was purchased (from year 3 to year 4).</p><p>c) Determine the increase in value of the card in the 20th year after it was purchased.</p>

<p>In 1990, a sum of $1000 is invested at a rate of 6% per year for 15 years.</p><p>Write an equation that models the growth of the investment, and use it to determine the value of the investment after 15 years.</p>

<p>In each case, write an equation that models the situation described. Explain what each part of each equation represents.</p><p> the population if a town had 2500 residents in 1990 and grew at a rate of
0.5% each year after that for 1‘ years</p>

<p>The temperature, <code class='latex inline'>T</code>, in degrees Celsius, of a cooling metal bar after t minutes is given by <code class='latex inline'>T(t) = 20 + 100(0.3)^{0.2t}</code>.</p><p>(a) Sketch the graph of this relation.</p><p>(b) What is the asymptote of this function? What does it represent?</p><p>(c) How long will it take for the temperature to be within <code class='latex inline'>0.1 ^{\circ}C</code> of the value of the asymptote?</p>

<p>A disinfectant is advertised as being able to kill 99% of all germs with each application.</p><p>a) Write an equation that represents the percent of germs left with
<code class='latex inline'>n</code> applications.</p><p>b) Suppose a kitchen countertop has 10 billion (<code class='latex inline'>10^{10}</code>) germs. How many
applications are required to eliminate all of the germs?</p>

<p>A computer, originally purchased for <code class='latex inline'>\$2000</code>, loses value according to the exponential equation <code class='latex inline'>V(t) = 2000(\frac{1}{2})^{\frac{t}{h}}</code>, where <code class='latex inline'>V</code> is the
value, in dollars, of the computer at any time,
<code class='latex inline'>t</code>, in years, after purchase and <code class='latex inline'>h</code> represents the half-life,in years, of the value of the computer. After 1 year, the computer has a value of approximately <code class='latex inline'>\$1516</code>.</p><p><strong>a)</strong> What is the half—life of the value of the
computer?</p><p><strong>b)</strong> How long will it take for the computer to be worth <code class='latex inline'>10\%</code> of its purchase price?</p>

<p>The equation that models the amount of time, t, in minutes that a cup of hot chocolate has been cooling as a function of its temperature, T, in degrees Celsius is <code class='latex inline'>t = \log(\frac{T - 22}{75}) \div \lgo(0.75)</code>. Calculate the following.</p><p>a) the cooling time if the temperature is 35<code class='latex inline'>^oC</code></p><p>b) the initial temperature of the drink.</p>

<p>The intensity of light energy under water decreases rapidly. Many factors affect how quickly the intensity decreases. The light energy under water can be calculated using an exponential relation. For example.</p>
<ul>
<li>Ocean: <code class='latex inline'>I= 325 \times (1.024)^{-d}</code></li>
<li>Lake Erie: <code class='latex inline'>I= 401 \times (1.222)^{-d}</code></li>
</ul>
<p>In these relations, <code class='latex inline'>d</code> is the depth in metres and <code class='latex inline'>I</code> is the light energy, in langleys.</p>
<ul>
<li>Why is a negative exponent used in the formulas?</li>
</ul>

<p>Use the decay equation for polonium-218, <code class='latex inline'>\displaystyle A(t) = A_0(\frac{1}{2})^{\frac{t}{3.1}}</code>, where <code class='latex inline'>A</code> is the amount remaining after t minutes and <code class='latex inline'>A_0</code> is the initial amount. </p>
<ul>
<li>How long will it take for this sample to decay to 10% of its initial amount of 50 mg?</li>
</ul>

<p>The number of bacteria in a bacterial culture is 2000 at 10:00 a.m. and doubles every hour. This can be expressed as
<code class='latex inline'> N = 2000 </code> x <code class='latex inline'>2^{t}</code>, where <em>N</em> represents the number of bacteria and <em>t</em> represents time, in hours. </p><p>a) Find the number of bacteria after 1 h. </p><p>b) Find the number of bacteria after 2 h. </p><p>c) Find the number of bacteria after 3 h. </p><p>d) Find the number of bacteria after 4 h. </p><p>e) What does <code class='latex inline'> t = 0</code> represent?</p><p>f) When were there 250 bacteria? Explain. </p>

<p>Chris walks halfway along a 100-m track in <code class='latex inline'>1</code> min. then half of the remaining distance in the next minute, then half of the remaining distance in the third minute, and so on.</p>
<ul>
<li>Will Chris get to the end of the track? Explain. Include a table of values and a graph to support your explanation.</li>
</ul>

<p>A population of yeast cells can double in as little as l h. Assume an initial population of 80 cells.</p><p>Approximately how many hours would it take for the population to
reach 1 million cells?</p>

<p>In 1990, a sum of $1000 is invested at a rate of 6% per year for 15 years.</p><p>How many growth periods are there?</p>

<p>Suppose a square-based pyramids a fixed height of 25 m.</p><p>What impact does doubling the volume have on the side length of the base? Explain.</p>

<p>The number of bees in a hive is 1000 on June 1 and doubles every month. This can be expressed as <code class='latex inline'>N=1000 \times 2^t</code>, where N represents the number of bees and t represents time, in months.</p><p>Find the number of bees after 2, 3, 4. and 5 months.</p>

<p>A radioactive substance with an initial mass of 100 mg has a half-life of 1.5 days.</p><p><strong>NOTE:</strong> To solve for x in exponential equation of <code class='latex inline'>a^x = b</code>, use <code class='latex inline'>x = \frac{\log b}{\log a}</code>.</p><p>(a) Write an equation to relate the mass remaining to time.</p><p>(b) Graph the function. Describe the shape of the curve.</p><p>(c) Limit the domain so that the model accurately describes the situation.</p><p>(d) Find the amount remaining after</p>
<ul>
<li>(i) 8 days</li>
<li>(ii) 2 weeks</li>
</ul>
<p>(e) How long will it take for the sample to decay to 3% of its initial mass?</p>

<p>When a patient takes a certain drug, <code class='latex inline'>\frac{1}{10}</code> of the drug that remains in his or her system is used per hour. A patient is given a 500 mg dose of a drug.</p><p>(a) Write an equation relating time and the remaining mass of the drug.</p>

<p>A population of yeast cells can double in as little as l h. Assume an initial population of 80 cells.</p><p>a) What is the growth rate, in percent per hour, of this colony ofyeast cells?</p><p>b) Write an equation that can be used to determine the population of cells
at <code class='latex inline'>t</code> hours.</p>

<p>A 20-mg sample of thorium-233 decays to 17 mg after 5 min.</p><p> Determine the half—life of thorium-233.</p>

<p>Television During one year, people in the United States older than 18 years old watched a total of 342 billion hours of television. The population of the United States older than 18 years old was about 209 million people.</p><p>a. On average, how many hours of television did each person older than 18 years old watch that year? Round to the nearest hour.</p><p>b. On average, how many hours per week did each person older than 18 years old watch that year? Round to the nearest hour.</p>

<p>Plans for Decimal Point call for a highway off-ramp to be built once the town’s population reaches 6500. When should the off—ramp be built?</p><p>Use <code class='latex inline'>
\displaystyle
P = 1006(1.016)^t
</code></p>

<p>The number of bees in a hive is 1000 on June 1 and doubles every month. This can be expressed as <code class='latex inline'>N=1000 \times 2^t</code>, where N represents the number of bees and t represents time, in months.</p>
<ul>
<li>What does <code class='latex inline'>t = 0</code> represent in this situation?</li>
</ul>

<p>For the given function find</p><p>a) whether it's Exponential Growth or Decay</p><p>b) its initial value</p><p>c) Growth or Decay Rate</p><p><code class='latex inline'>\displaystyle
V(t) = 20*(1.02)^t
</code></p>

<p>A radioactive sample with an initial mass of 25 mg has a half-life of 2 days.</p><p>(a) Which equation models this exponential decay, where t is the time, in days, and A is the amount of the substance that remains?</p>
<ul>
<li>(A) <code class='latex inline'>A = 25 \times 2^{\frac{t}{2}}</code></li>
<li>(B) <code class='latex inline'>A = 25 \times (\frac{1}{2})^{2t}</code></li>
<li>(C) <code class='latex inline'>A = 25 \times (\frac{1}{2})^{\frac{t}{2}}</code></li>
<li>(D) <code class='latex inline'>A = 2 \times25^{\frac{t}{2}}</code></li>
</ul>
<p>(b) What is the amount of radioactive material remaining after 7 days?</p>

<p>A 10-mg sample of bismuth-214 decays to 9 mg in 3 min.</p>
<ul>
<li>Graph the amount of bismuth—214 remaining as a function of time.</li>
</ul>

<p>A group of yeast cells grows by 75% every 3 h. At 9 a.m., there are 200 yeast cells.</p><p>a) Write an equation that models the number of cells, given the number of hours after 9 am.</p><p>b) Explain how each part of your equation is related to the given information.</p>

<p>Use the decay equation for polonium-218, <code class='latex inline'>\displaystyle A(t) = A_0(\frac{1}{2})^{\frac{t}{3.1}}</code>, where <code class='latex inline'>A</code> is the amount remaining after t minutes and <code class='latex inline'>A_0</code> is the initial amount. </p>
<ul>
<li>How much will remain after 90 s from an initial sample of 50 mg?</li>
</ul>

<p>Half-life is the time it takes for half of a sample of a radioactive element to decay. The function <code class='latex inline'>\displaystyle{M(t)=P\left(\frac{1}{2}\right)^{\frac{t}{h}}}</code> can be used to calculate the mass remaining if the half-life is <code class='latex inline'>h</code> and the initial mass is <code class='latex inline'>P</code>. The half-life of radium is 1620 years.</p>
<ul>
<li>If a laboratory has 5 g of radium, how much will there be in 150 years?</li>
</ul>

<p>The intensity of light energy under water decreases rapidly. Many factors affect how quickly the intensity decreases. The light energy under water can be calculated using an exponential relation. For example.</p>
<ul>
<li>Ocean: <code class='latex inline'>I= 325 \times (1.024)^{-d}</code></li>
<li>Lake Erie: <code class='latex inline'>I= 401 \times (1.222)^{-d}</code></li>
</ul>
<p>In these relations, <code class='latex inline'>d</code> is the depth in metres and <code class='latex inline'>I</code> is the light energy, in langleys.</p>
<ul>
<li>Sketch a graph of each relation.</li>
</ul>

<p>A particular radioactive substance has a half-life of 3 years. Suppose an initial sample has a mass of 200 mg. </p><p><strong>NOTE:</strong> To solve for x in exponential equation of <code class='latex inline'>a^x = b</code>, use <code class='latex inline'>x = \frac{\log b}{\log a}</code>.</p><p>(a) Write the equation that relates the mass of radioactive material remaining to time.</p><p>(b) How much will remain after one decade?</p><p>(c) How long will it take for the sample to decay to 10% of its initial mass? Explain how you arrived at your answer.</p>

<p>A student records the internal temperature of a hot sandwich that has been left to cool on a kitchen counter. The room temperature is 19 <code class='latex inline'>^oC</code>. An equation that models this situation is</p><p><code class='latex inline'>\displaystyle
T(t) = 63(0.5)^{\frac{t}{10}} +9
</code></p><p>where Tis the temperature in degrees Celsius and t is the time in minutes.</p><p> What was the temperature of the sandwich when she began to record
its temperature?</p>

<p>Solve the exponential equations.</p><p><code class='latex inline'>\displaystyle
500 = N_0(1.25)^{1.25}
</code></p>

<p> Carbon- 14 is present in all living organisms and decays at a predictable rate. To estimate the age of an organism, archaeologists measure the amount of carbon- 14 left in its remains. The approximate amount of carbon-14 remaining after 5000 years can be found using the formula <code class='latex inline'>\displaystyle A=A_{0}(2.7)^{-\frac{3}{5}} </code>, where <code class='latex inline'>\displaystyle A_{0} </code> is the initial amount of carbon-14 in the sample that is tested. How much carbon- 14 is left in a sample that is 5000 years old and originally contained <code class='latex inline'>\displaystyle 7.0 \times 10^{-12} </code> grams of carbon- <code class='latex inline'>\displaystyle 14 ? </code></p>