1 WE30 Changing δ-gluconolactone into gluconic acid can be modelled by the equation y = y0e-0.6t, where y is the number of grams of δ-gluconolactone present t hours after the process has begun. Suppose 200 grams of δ-gluconolactone is to be changed into gluconic acid.
a Find the value of y0.
b Write the equation replacing y0 with your answer.
c How many grams of δ-gluconolactone will be present after 1 hour? Give your answer correct to the nearest gram.
d How long will it take to reduce the amount of δ-gluconolactone to 50 grams? Give your answer correct to the nearest quarter of an hour.
2 The decay of radon-222 gas is given by the equation y = y0e-0.18t, where y is the amount of radon remaining after t days. When t = 0, y = 10 g. Give all answers to the nearest whole number.
a Find the value of y0.
b Write the equation substituting your value of y0. c What will be the mass after 1 day?
d How many days will it take for the mass to reach 1 g?
3 The equation y = A + B loge (x) relates two variables x and y. The table below shows values of x and y.
x 1 2 3
y 3 4.386 m
a Find the value of A and B correct to the nearest whole number.
b Write the equation relating x and y substituting values for A and B.
c Using your new equation, find the value of m correct to 3 decimal places.
d If y = 7.6, find x correct to the nearest whole number.
ExErCisE
3i
4 An amount of $1000 is invested in a building society where the 5% p.a. interest paid is compounded continuously. The amount in the account after t years can be modelled by the equation A = A0ert, where r is the continuous interest rate.
a Find the value of A0 and r.
b Write the equation substituting values of A0 and r.
c Find the amount in the bank after i 1 year ii 10 years. Give your answer correct to the nearest dollar.
d How long will it take for the investment to double in value? Give your answer to the nearest year.
5 The number of people living in Boomerville at any time, t years, after the first settlers arrived can be modelled by the equation P = P0ekt. Suppose 500 people arrived on 1 January 1850, and by 1 January 1860 there were 675 people.
a What is the value of P0?
b Find the value of k correct to 2 decimal places.
c Write the equation substituting values for P0 and k.
d What will be the population on 1 January 1900? Give your answer to the nearest 10 people.
e When will the population be 2000?
6 A cup of soup cools to the temperature of the surrounding air. Newton’s Law of Cooling can be written as T - TS= (T0- TS)e-kt, where T is the temperature of the object after t minutes, and TS is the temperature of the surrounding air. The soup cooled from 90 °C to 70 °C after 6 minutes in a room with an air temperature of 15 °C.
a Find the values of TS, T0 and k correct to 2 decimal places.
b Write the equation substituting the values for TS, T0 and k.
c Find the temperature of the soup after 10 minutes. Give your answer to the nearest degree.
d How long would it take for the soup to be 40 °C? Give your answer to the nearest minute.
e If the soup is placed in a refrigerator in which the temperature is 2 °C, how long will it take for the soup to reach 40 °C? Use the same value of k and give your answer to the nearest minute.
7 The diameter of a tree for a period of its growth can be modelled by the equation D = D0ekt, where t is the number of years after the beginning of the period. The diameter of the tree grew from 50 cm to 60 cm in the first 2 years that measurements were taken.
a Find the values of D0 and k.
b Write the equation using these values.
c How much will it have grown in the first 5 years? Round to the nearest centimetre.
d How long will it take for the tree’s diameter to double? Round to the nearest year.
8 The decay of a radioactive substance can be modelled by the equation M = M0e-kt, where M grams is the mass of the substance after t years. After 10 years the mass of the substance is 98 grams and after 20 years the mass is 96 grams.
a What was the mass of the substance initially? Give your answer to the nearest gram.
b Find the value of k. Give your answer to 3 decimal places.
c Write the equation using these values.
d Find the mass of the substance after 50 years.
e How long would it take for the mass to be halved?
9 The number of bacteria present in a culture at any time, t hours, can be modelled by the equation N = N0ekt.
a If the original number is doubled in 3 hours, find k correct to 2 decimal places.
b Write the equation substituting the value of k.
c Find the original number of bacteria if there were 2500 bacteria after 4 hours. Give the answer correct to the nearest thousand.
d Write the equation substituting your value for the original population.
e Find the number of bacteria present after 8 hours. Give your answer correct to the nearest thousand.
10 The intensity of light d metres below the surface of the sea can be modelled by the equation I = I0e-kd. Divers in the Sea of Loga have found that the intensity of light is halved when a diver is 5 metres below the surface of the water.
a Find the value of k correct to 4 decimal places.
b Write the equation substituting the value of k.
c Find the percentage of light available at a depth of 10 metres.
d If artificial light is necessary when the intensity of light is less than 0.1 of the intensity at the surface (I < 0.1I0), find how deep a diver can go before artificial light is necessary.
summary
Index laws a
• x× ay= ax + y • ax÷ ay= ax - y (
• ax)y= axy • a0= 1
a a
x x
− = 1
• and 1
a−x =ax ay1 =ya
• and axy =yax a
• x= y ⇔ loga (y) = x Logarithm laws log
• a (1) = 0 log
• a (a) = 1 log
• a (0) is undefined.
log
• a (mn) = loga (m) + loga (n) loga m log ( ) log ( )a a
n m n
= −
• log
• a (mp) = p loga (m) log ( ) log ( )
log ( )
b a
a
N N
= b
•
Exponential equations
To solve exponential equations:
•
write all terms with the same base, write terms with the smallest possible base or take the logarithm of 1.
both sides of the equation then solve the equation.
2.
A negative number cannot be expressed in index form.
• If 0
• < x < 1, then loga (x) < 0 and if x > 1 then loga (x) > 0.
It is not possible to take the logarithm of a negative number.
•
Exponential equations (base e) Euler’s number
• e
n
h
n
= +
=
lim→∞ 1 1 . ...
2 718 281828459
The laws of indices and logarithms apply in the same way when using
• e.
Equations with natural (base e) logarithms
To solve logarithmic equations use the laws of logarithms and indices.
•
Inverses The equation
• y = loge (x) is an inverse function of y = ex. To find an inverse, interchange
• x and y.
alog ( )a x =x x, ∈R+
•
log ( )a ax =x x R, ∈
•
Literal equations An equation such as
• ekx= a, where k ∈ R and a ∈ R+, is called a literal equation.
Literal equations do not have numerical solutions.
•
The solution of a literal equation is expressed in terms of the other pronumerals, in this case
• a and k, often
called parameters.