91Ó°ÊÓ

Simionie needs S7000 to buy a snowmobile, but only has \(\$ 6000 .\) His bank offers a GIC that pays an annual interest rate of \(3.93 \%,\) compounded annually. How long would Simionie have to invest his money in the GIC to have enough money to buy the snowmobile?

A biologist places agar, a gel made from seaweed, in a Petri dish and infects it with bacteria. She uses the measurement of the growth ring to estimate the number of bacteria present. The biologist finds that the bacteria increase in population at an exponential rate of \(20 \%\) every 2 days. a) If the culture starts with a population of 5000 bacteria, what is the transformed exponential function in the form \(P=a(c)^{b x}\) that represents the population, \(P,\) of the bacteria over time, \(x,\) in days? b) Describe the parameters used to create the transformed exponential function. c) Graph the transformed function and use it to predict the bacteria population after 9 days.

A \(\$ 1000\) investment earns interest at a rate of \(8 \%\) per year, compounded quarterly. a) Write an equation for the value of the investment as a function of time, in years. b) Determine the value of the investment after 4 years. c) How long will it take for the investment to double in value?

Money in a savings account earns compound interest at a rate of \(1.75 \%\) per year. The amount, \(A,\) of money in an account can be modelled by the exponential function \(A=P(1.0175)^{n}\) where \(P\) is the amount of money first deposited into the savings account and \(n\) is the number of years the money remains in the account. a) Graph this function using a value of \(P=\$ 1\) as the initial deposit. b) Approximately how long will it take for the deposit to triple in value? c) Does the amount of time it takes for a deposit to triple depend on the value of the initial deposit? Explain. d) In finance, the rule of 72 is a method of estimating an investment's doubling time when interest is compounded annually. The number 72 is divided by the annual interest rate to obtain the approximate number of years required for doubling. Use your graph and the rule of 72 to approximate the doubling time for this investment.

Cobalt-60 (Co-60) has a half-life of approximately 5.3 years. a) Write an exponential function to model this situation. b) What fraction of a sample of Co-60 will remain after 26.5 years? c) How long will it take for a sample of \(\mathrm{Co}-60\) to decay to \(\frac{1}{512}\) of its original mass?

Statistics indicate that the world population since 1995 has been growing at a rate of about \(1.27 \%\) per year. United Nations records estimate that the world population in 2011 was approximately 7 billion. Assuming the same exponential growth rate, when will the population of the world be 9 billion?

a) On the same set of axes, sketch the graph of the function \(y=5^{x},\) and then sketch the graph of the inverse of the function by reflecting its graph in the line \(y=x\) b) How do the characteristics of the graph of the inverse of the function relate to the characteristics of the graph of the original exponential function? c) Express the equation of the inverse of the exponential function in terms of \(y\) That is, write \(x=F(y)\)

The Krumbein phi scale is used in geology to classify sediments such as silt, sand, and gravel by particle size. The scale is modelled by the function \(D(\varphi)=2^{-\varphi}\) where \(D\) is the diameter of the particle, in millimetres, and \(\varphi\) is the Krumbein scale value. Fine sand has a Krumbein scale value of approximately \(3 .\) Coarse gravel has a Krumbein scale value of approximately -5 a) Why would a coarse material have a negative scale value? b) How does the diameter of fine sand compare with the diameter of coarse gravel?

The formula for calculating the monthly mortgage payment, \(P M T,\) for a property is \(P M T=P V\left[\frac{i}{1-(1+i)^{-n}}\right],\) where \(P V\) is the present value of the mortgage; \(i\) is the interest rate per compounding period, as a decimal; and \(n\) is the number of payment periods. To buy a house, Tyseer takes out a mortgage worth \(\$ 150\) 000 at an equivalent monthly interest rate of \(0.25 \%\) He can afford monthly mortgage payments of \(\$ 831.90 .\) Assuming the interest rate and monthly payments stay the same, how long will it take Tyseer to pay off the mortgage?