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>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>Vanessa, the student council president, needed to get a message to the whole school, but she did not have time to e-mail every student. So she set up an e-mail tree. She sent the message to her two vice-presidents, and asked them each to forward it to two students. Suppose that this pattern is repeated and assume that no one receives the e-mail more than once.</p><p>(b) On which mailing will the e-mail be sent to at least 500 people? Explain how you found this answer.</p><p>(c) At this point, how many people would have received the e-mail, in total?</p><p>(d) There are 500 students in the school. How many mailings would it take to reach all the students?</p>

<p>A piece of wood burns completely in <code class='latex inline'>1</code> s
at 600°C. The time it takes for the wood to burn is doubled for every 10°C drop in A temperature and halved for every 10<code class='latex inline'>^oC</code> increase in temperature. Determine how long the piece of wood would take to burn‘ at each temperature.</p><p>a) <code class='latex inline'>500^oC</code></p><p>b) <code class='latex inline'>650^oC</code></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>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>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 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>Write the equation that models each situation. In each case, describe each part ofyour equation.</p><p>a) the percent of a pond covered by water lilies if they cover one—third of a pond now and each week they increase their coverage by 10% </p><p>b) the amount remaining of the radioactive isotope U238 if it has a half—life of <code class='latex inline'>4.5 \times 10^9</code> years</p><p>c) the intensity of light if each gel used to change the colour of a spotlight reduces the intensity of the light by 4%</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>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>Vanessa, the student council president, needed to get a message to the whole school, but she did not have time to e-mail every student. So she set up an e-mail tree. She sent the message to her two vice-presidents, and asked them each to forward it to two students. Suppose that this pattern is repeated and assume that no one receives the e-mail more than once.</p><p>a) How many people will receive the e-mail on the seventh mailing?</p>

<p>The battery power available to operate a deep space probe is given by the formula <code class='latex inline'>\displaystyle P=42 e^{-0.005 t} </code>, where <code class='latex inline'>\displaystyle P </code> is power in watts and <code class='latex inline'>\displaystyle t </code> is time in years. For how many years can the probe run if it requires 35 watts? Round the answer to the nearest tenth year.</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 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>a) Build a volume model to represent a cube with side length 3 cm. Sketch the model and label the length, width, and height.</p><p>b) What is the volume of the cube? Write this as a power.</p><p>c) Write an expression for the area of one face of the cube as a power. Evaluate the area of one face.</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>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>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>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>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>WILDLIFE A population of 100 deer is reintroduced to a wildlife preserve. Suppose the population does extremely well and the deer population doubles in two years. Then the number <code class='latex inline'>\displaystyle D </code> of deer after <code class='latex inline'>\displaystyle t </code> years is given by <code class='latex inline'>\displaystyle D=100 \cdot 2^{\frac{t}{2}} </code>.</p><p>a. How many deer will there be after <code class='latex inline'>\displaystyle 4 \frac{1}{2} </code> years?</p><p>b. Make a table that charts the population of deer every year for the next five years.</p><p>c. Make a graph using your table.</p><p>d. Using your table and graph, decide whether this is a reasonable trend over the long term. Explain.</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>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>Cobalt—60 is a radioactive element that is used to sterilize medical equipment.</p><p>Cobalt-60 decays to <code class='latex inline'>\frac{1}{2}</code>, or <code class='latex inline'>2^{-1}</code>. of its original amount after every 5.2 years. Determine the remaining mass of 20 g of cobalt-60 after</p><p><strong>a)</strong> 20.8 years </p><p><strong>b)</strong> 36.4 years</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>Systems of equations can involve non-linear relations. Consider
the population growth patterns of two towns since the turn of the
century.</p><img src="/qimages/1217" />
<ul>
<li>Identify the solution of this system, and explain what it means.</li>
</ul>

<p>The population of a city is growing at an average rate of <code class='latex inline'>3\%</code> per year. In 1990, the population was 45 000. </p><p>a) Write an equation that models the growth of the city. Explain what each part of the equation represents.</p><p>b) Use your equation to determine the population of the city in 2007.</p><p>c) Determine the year during which the population will have doubled.</p><p>d) Suppose the population took only 10 years to double. What growth rate would be required for this to have happened?</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>Mary must solve the equation <code class='latex inline'>\displaystyle 1.225=(1+i)^{12} </code> to determine the value of each dollar she invested for a year at the interest rate <code class='latex inline'>\displaystyle i </code> per year. Her friend Bindu suggests that she begin by taking the 12 th root of each side of the equation. Will this work? Try it and solve for the variable <code class='latex inline'>\displaystyle i </code>. Explain why it does or does not work.</p>

<p>E. coli is a type of bacteria that lives in our intestines and is necessary for digestion. It doubles in population every 20 min. The initial population is 10.</p><p>a) Copy and complete the table. </p><img src="/qimages/21800" /><p>b) When will the population of E. coli overtake the population of Listeria?</p><p>c) What population will the two cultures have when they are equal?</p>

<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 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>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> In 2008 , there were about <code class='latex inline'>\displaystyle 1.5 </code> billion Internet users. That number is projected to grow to <code class='latex inline'>\displaystyle 3.5 </code> billion in 2015 .</p><p>a. Let <code class='latex inline'>\displaystyle t </code> represent the time, in years, since 2008 . Write a function of the form <code class='latex inline'>\displaystyle y=a e^{c t} </code> that models the expected growth in the population of Internet users.</p><p>b)In what year are there 2 billion Internet users?
c) In what year are there 5 billion Internet users?
d)Solve your equation for <code class='latex inline'>\displaystyle t </code>.<br>
e)Explain how you can use your equation from part answers to parts (b) and (c).</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>Listeria is a type of bacteria that can cause dangerous health problems. It doubles every hour. The initial population of a sample of Listeria is 800.</p><p>a) Copy and complete this table, which shows the population of Listeria over time. </p><img src="/qimages/21801" /><p>b) Construct a graph of population versus time. Use a smooth curve to connect the points. Describe the shape of the graph.</p><p>c) What will the population be after </p>
<ul>
<li>1 day?</li>
<li>2 days?</li>
</ul>
<p>d) The symptoms of food poisoning can start as quickly as 4 h after eating contaminated food or as long as 24 h later. Discuss why some types of food poisoning begin quickly and other much more slowly.</p>

<p>Systems of equations can involve non-linear relations. Consider
the population growth patterns of two towns since the turn of the
century.</p>
<ul>
<li>Numberton has been growing steadily by 1000 every year, from
an initial population of 25 000.</li>
<li>Decimalville has been growing by 10% of its previous year's
population every year, from an initial population of 15 000.</li>
</ul>
<img src="/qimages/1217" /><p>Copy and complete the table of values up to 15 years.
Round to the nearest whole number if necessary.</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>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>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>Uranium is a radioactive material that emits energy when it changes into another substance. Uranium comes in different forms, called isotopes. One isotope, U-235, has a half-life, of 23 min, which means that it takes 23 min for a sample to decay to half its original amount. </p><p>a) Suppose you started with a 100 mg sample of U-235. </p><img src="/qimages/21799" /><p>b) Construct a graph of the amount, in milligrams, of U-235 remaining versus time, in minutes. Describe the shape of the graph.</p><p>c) Approximately how much U-235 will remain after 2h?</p><p>d) How long will it take until only 1 mg of U-235 remains?</p><p>e) Use the pattern in the table to write an expression, using powers of \frac{1}{2}, for the original amount of U-235. Does this make sense?</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>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>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>E. coli is a type of bacteria that can cause dangerous health problems. It doubles every 20 min. The initial population of a sample of E. coli is 400.</p><p>a) Copy and complete this table.</p><p><code class='latex inline'>\displaystyle
\begin{array}{|c|c|}
\hline Time (min) & \text{Population of E. coli} \\
\hline 0 & 400 \\
\hline 20 & 800 \\
\hline 40 & \\
\hline 60 & \\
\hline 80 & \\
\hline 100 & \\
\hline 120 & \\
\hline
\end{array}
</code></p><p>b) Construct a graph of population versus time. Use a smooth curve to connect the points. Describe the shape of the graph.</p><p>c) What will the population be after</p>
<ul>
<li><p>5 h?</p></li>
<li><p>8 h?</p></li>
</ul>

<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>The duration (lengths of time) of musical notes are related by powers of <code class='latex inline'>\frac{1}{2}</code>, beginning with a whole note. </p><img src="/qimages/21798" />

<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>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>Investing The expression <code class='latex inline'>1000(1.1)^t</code> represents the value of a $1000 investment that earns 10% interest per year, compounded annually for <code>t$</code> years. What is the value of a $1000 investment at the end of each period?</p><p>24 years</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>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>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>Joan won a multi-million dollar lottery. She decides to give $1 000 000 of her winnings to charity. Her plan is to give <code class='latex inline'>\frac{1}{2}</code> or <code class='latex inline'>2^{-1}</code>, to charity in January, and then give half of the remaining amount in February, half again in March, and so on.</p><p>a) What fraction remains after 6 months?</p><p>b) What fraction remains after 12 months?</p><p>c) Write each fraction as a power of 2 with a negative exponent.</p><p>d) What amount is remaining at the end of the year?</p>

<p>Systems of equations can involve non-linear relations. Consider
the population growth patterns of two towns since the turn of the
century.</p><img src="/qimages/1217" /><p>i) Graph population versus years for the towns on the same grid.</p><p>ii) Classify each relation as linear or non-linear.</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>(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> 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>

<p>The half-life of carbon-14 (C-14) is 5700 years.</p><p>a) Copy and complete the table for a 50-g sample of C-14.</p><img src="/qimages/6905" /><p>b) Construct a graph of the amount of C-14 remaining versus time, in years. Describe the shape of the graph.</p><p>c) Approximately how much C-14 will remain after 20 000 years?</p><p>d) How long will it take until only 1 g of C-14 remains?</p>