Exponential Functions

6 In a chemical reaction, the mass \(m\) grams of a chemical after \(t\) minutes is modelled by the equation $$m = 20 + 30 \mathrm { e } ^ { - 0.1 t }$$

1. Find the initial mass of the chemical. What is the mass of chemical in the long term?
2. Find the time when the mass is 30 grams.
3. Sketch the graph of \(m\) against \(t\).

6. In a chemical reaction, the mass \(m\) grams of a chemical after \(t\) minutes is modelled by the equation $$m = 20 + 30 \mathrm { e } ^ { - 0.1 t }$$

1. Find the initial mass of the chemical. What is the mass of chemical in the long term?
2. Find the time when the mass is 30 grams.
3. Sketch the graph of \(m\) against \(t\).

8. \begin{figure}[h] \end{figure} The value of Lin's car is modelled by the formula $$V = 18000 \mathrm { e } ^ { - 0.2 t } + 4000 \mathrm { e } ^ { - 0.1 t } + 1000 , \quad t \geqslant 0$$ where the value of the car is \(V\) pounds when the age of the car is \(t\) years.
A sketch of \(t\) against \(V\) is shown in Figure 1.

1. State the range of \(V\). According to this model,
2. find the rate at which the value of the car is decreasing when \(t = 10\) Give your answer in pounds per year.
3. Calculate the exact value of \(t\) when \(V = 15000\)

4 The graph of \(y = a b ^ { x }\) passes through the points \(( 1,6 )\) and \(( 2,3.6 )\). Find the values of \(a\) and \(b\).

3 It is given that \(x\) is proportional to the product of the square of \(y\) and the positive square root of \(z\). When \(y = 2\) and \(z = 9 , x = 30\).

1. Write an equation for \(x\) in terms of \(y\) and \(z\).
2. Find the value of \(x\) when \(y = 3\) and \(z = 25\).

7

1. Substance \(A\) is decaying exponentially and its mass is recorded at regular intervals. At time \(t\) years, the mass, \(M\) grams, of substance \(A\) is given by $$M = 40 \mathrm { e } ^ { - 0.132 t }$$ (a) Find the time taken for the mass of substance \(A\) to decrease to \(25 \%\) of its value when \(t = 0\).
   (b) Find the rate at which the mass of substance \(A\) is decreasing when \(t = 5\).
2. Substance \(B\) is also decaying exponentially. Initially its mass was 40 grams and, two years later, its mass is 31.4 grams. Find the mass of substance \(B\) after a further year.

9 The points \(( 2,6 )\) and \(( 3,18 )\) lie on the curve \(y = a x ^ { n }\).
Use logarithms to find the values of \(a\) and \(n\), giving your answers correct to 2 decimal places.

1. The mass, \(A\) kg, of algae in a small pond, is modelled by the equation

$$A = p q ^ { t }$$ where \(p\) and \(q\) are constants and \(t\) is the number of weeks after the mass of algae was first recorded. Data recorded indicates that there is a linear relationship between \(t\) and \(\log _ { 10 } A\) given by the equation $$\log _ { 10 } A = 0.03 t + 0.5$$

1. Use this relationship to find a complete equation for the model in the form $$A = p q ^ { t }$$ giving the value of \(p\) and the value of \(q\) each to 4 significant figures.
2. With reference to the model, interpret
   1. the value of the constant \(p\),
   2. the value of the constant \(q\).
3. Find, according to the model,
   1. the mass of algae in the pond when \(t = 8\), giving your answer to the nearest 0.5 kg ,
   2. the number of weeks it takes for the mass of algae in the pond to reach 4 kg .
4. State one reason why this may not be a realistic model in the long term.

10 A student conducts an experiment and records the following data for two variables, \(x\) and \(y\). The student is told that the relationship between \(x\) and \(y\) can be modelled by an equation of the form \(y = k b ^ { x }\) 10

1. Plot values of \(\log _ { 10 } y\) against \(x\) on the grid below.
   [0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{3176ee0c-fba2-4878-af3a-c3ac092bbc1f-10_1086_1205_1037_536} 10
2. State, with a reason, which value of \(y\) is likely to have been recorded incorrectly.
   [0pt] [1 mark] 10
3. By drawing an appropriate straight line, find the values of \(k\) and \(b\).
   [0pt] [4 marks]

1. (a) Sketch the curve \(y = 5 ^ { x - 1 }\), showing the coordinates of any points of intersection with the coordinate axes.
   (b) Find, to 3 significant figures, the \(x\)-coordinates of the points where the curve \(y = 5 ^ { x - 1 }\) intersects
   1. the straight line \(y = 10\),
   2. the curve \(y = 2 ^ { x }\).
   $$f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 6 x + 1 .$$

8 \includegraphics[max width=\textwidth, alt={}, center]{b2c1188d-881e-4fb5-bece-5a51006543c7-4_524_822_274_609} The diagram shows the curves \(y = a ^ { x }\) and \(y = 4 b ^ { x }\).

1. (a) State the coordinates of the point of intersection of \(y = a ^ { x }\) with the \(y\)-axis.
   (b) State the coordinates of the point of intersection of \(y = 4 b ^ { x }\) with the \(y\)-axis.
   (c) State a possible value for \(a\) and a possible value for \(b\).
2. It is now given that \(a b = 2\). Show that the \(x\)-coordinate of the point of intersection of \(y = a ^ { x }\) and \(y = 4 b ^ { x }\) can be written as $$x = \frac { 2 } { 2 \log _ { 2 } a - 1 } .$$

9

1. Sketch the graph of \(y = 4 k ^ { x }\), where \(k\) is a constant such that \(k > 1\). State the coordinates of any points of intersection with the axes.
2. The point \(P\) on the curve \(y = 4 k ^ { x }\) has its \(y\)-coordinate equal to \(20 k ^ { 2 }\). Show that the \(x\)-coordinate of \(P\) may be written as \(2 + \log _ { k } 5\).
3. (a) Use the trapezium rule, with two strips each of width \(\frac { 1 } { 2 }\), to find an expression for the approximate value of $$\int _ { 0 } ^ { 1 } 4 k ^ { x } \mathrm {~d} x$$ (b) Given that this approximate value is equal to 16 , find the value of \(k\).

1 Solve the equation \(\mathrm { e } ^ { 2 x } - 5 \mathrm { e } ^ { x } = 0\).

1 Solve the equation \(\mathrm { e } ^ { 2 x } - 5 \mathrm { e } ^ { x } = 0\).

2 Using the substitution \(u = 3 ^ { x }\), solve the equation \(3 ^ { x } + 3 ^ { 2 x } = 3 ^ { 3 x }\) giving your answer correct to 3 significant figures.

2 A zoologist is investigating the growth of a population of red squirrels in a forest.
She uses the equation \(N = \frac { 200 } { 1 + 9 \mathrm { e } ^ { - \frac { t } { 5 } } }\) as a model to predict the number of squirrels, \(N\), in the population \(t\) weeks after the start of the investigation. What is the size of the squirrel population at the start of the investigation?
Circle your answer.
5
20
40
200

2 A zoologist is investigating the growth of a population of red squirrels in a forest.
She uses the equation \(N = \frac { 200 } { 1 + 9 \mathrm { e } ^ { - \frac { t } { 5 } } }\) as a model to predict the number of squirrels, \(N\), in the population \(t\) weeks after the start of the investigation. What is the size of the squirrel population at the start of the investigation?
Circle your answer.
5
20
40
200

4. A population of deer is introduced into a park. The population \(P\) at \(t\) years after the deer have been introduced is modelled by \(P = \frac { 2000 a ^ { t } } { 4 + a ^ { t } }\), where \(a\) is a constant. Given that there are 800 deer in the park after 6 years,

1. calculate, to 4 decimal places, the value of \(a\),
2. use the model to predict the number of years needed for the population of deer to increase from 800 to 1800.
3. With reference to this model, give a reason why the population of deer cannot exceed 2000.

5. Solve

1. \(2 ^ { y } = 8\)
2. \(2 ^ { x } \times 4 ^ { x + 1 } = 8\)

5. Solve

1. \(2 ^ { y } = 8\)
2. \(2 ^ { x } \times 4 ^ { x + 1 } = 8\)

2 Solve the equation $$5 ^ { x - 1 } = 5 ^ { x } - 5$$ giving your answer correct to 3 significant figures.

6

1. Find \(\sum _ { k = 2 } ^ { 5 } 2 ^ { k }\).
2. Find the value of \(n\) for which \(2 ^ { n } = \frac { 1 } { 64 }\).
3. Sketch the curve with equation \(y = 2 ^ { x }\).

2 The graph of \(y = 5 ^ { x }\) is transformed by a stretch in the \(y\)-direction, scale factor 5 State the equation of the transformed graph. Circle your answer.
[0pt] [1 mark] \(y = 5 \times 5 ^ { x }\) \(y = 5 ^ { \frac { x } { 5 } }\) \(y = \frac { 1 } { 5 } \times 5 ^ { x }\) \(y = 5 ^ { 5 x }\)

14

1. 1. Given that $$y = 2 ^ { x }$$ write down \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) 14
2. (ii) Hence find $$\int 2 ^ { x } \mathrm {~d} x$$ 14
3. The area, \(A\), bounded by the curve with equation \(y = 2 ^ { x }\), the \(x\)-axis, the \(y\)-axis and the line \(x = - 4\) is approximated using eight rectangles of equal width as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-23_1319_978_450_532} 14
4. 1. Show that the exact area of the largest rectangle is \(\frac { \sqrt { 2 } } { 4 }\) 14
5. (ii) The areas of these rectangles form a geometric sequence with common ratio \(\frac { \sqrt { 2 } } { 2 }\) Find the exact value of the total area of the eight rectangles.
   Give your answer in the form \(k ( 1 + \sqrt { 2 } )\) where \(k\) is a rational number.
   [0pt] [3 marks]
   14
6. (iii) More accurate approximations for \(A\) can be found by increasing the number, \(n\), of rectangles used. Find the exact value of the limit of the approximations for \(A\) as \(n \rightarrow \infty\)

5 A substance is decaying in such a way that its mass, m kg , at a time t years from now is given by the formula $$\mathrm { m } = 240 \mathrm { e } ^ { - 0.04 \mathrm { t } }$$

1. Find the time taken for the substance to halve its mass.
2. Find the value of \(t\) for which the mass is decreasing at a rate of 2.1 kg per year.

6 The mass \(M \mathrm {~kg}\) of a radioactive material is modelled by the equation $$M = M _ { 0 } \mathrm { e } ^ { - k t } ,$$ where \(M _ { 0 }\) is the initial mass, \(t\) is the time in years, and \(k\) is a constant which measures the rate of radioactive decay.

1. Sketch the graph of \(M\) against \(t\).
2. For Carbon \(14 , k = 0.000121\). Verify that after 5730 years the mass \(M\) has reduced to approximately half the initial mass. The half-life of a radioactive material is the time taken for its mass to reduce to exactly half the initial mass.
3. Show that, in general, the half-life \(T\) is given by \(T = \frac { \ln 2 } { k }\).
4. Hence find the half-life of Plutonium 239, given that for this material \(k = 2.88 \times 10 ^ { - 5 }\).

9. A scientist is using carbon-14 dating to determine the age of some wooden items. The equation for carbon-14 dating an item is given by $$N = k \lambda ^ { t }$$ where

• \(N\) grams is the amount of carbon-14 currently present in the item
• \(k\) grams was the initial amount of carbon-14 present in the item
• \(t\) is the number of years since the item was made
• \(\lambda\) is a constant, with \(0 < \lambda < 1\)
  1. Sketch the graph of \(N\) against \(t\) for \(k = 1\)

Given that it takes 5700 years for the amount of carbon-14 to reduce to half its initial value,

show that the value of the constant \(\lambda\) is 0.999878 to 6 decimal places. Given that Item \(A\)

• is known to have had 15 grams of carbon-14 present initially
• is thought to be 3250 years old
• calculate, to 3 significant figures, how much carbon-14 the equation predicts is currently in Item \(A\).

Item \(B\) is known to have initially had 25 grams of carbon-14 present, but only 18 grams now remain.

Use algebra to calculate the age of Item \(B\) to the nearest 100 years.

9 An analyst believes that the sales of a particular electronic device are growing exponentially. In 2015 the sales were 3.1 million devices and the rate of increase in the annual sales is 0.8 million devices per year.

1. Find a model to represent the annual sales, defining any variables used.
2. In 2017 the sales were 5.2 million devices. Determine whether this is consistent with the model in part (i).
3. The analyst uses the model in part (i) to predict the sales for 2025. Comment on the reliability of this prediction.

6 A circle has equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 4 y - 71 = 0$$ 6

1. Find the centre of the circle.
   6
2. Hence, find the equation of the tangent to the circle at the point (1, 10), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

5. The mass, \(m\) grams, of a leaf \(t\) days after it has been picked from a tree is given by $$m = p \mathrm { e } ^ { - k t }$$ where \(k\) and \(p\) are positive constants.
When the leaf is picked from the tree, its mass is 7.5 grams and 4 days later its mass is 2.5 grams.

1. Write down the value of \(p\).
2. Show that \(k = \frac { 1 } { 4 } \ln 3\).
3. Find the value of \(t\) when \(\frac { \mathrm { d } m } { \mathrm {~d} t } = - 0.6 \ln 3\).

3 The mass, \(m\) grams, of a substance at time \(t\) years is given by the formula $$m = 180 \mathrm { e } ^ { - 0.017 t }$$

1. Find the value of \(t\) for which the mass is 25 grams.
2. Find the rate at which the mass is decreasing when \(t = 55\).

3 The mass, \(m\) grams, of a substance at time \(t\) years is given by the formula $$m = 180 \mathrm { e } ^ { - 0.017 t } .$$

1. Find the value of \(t\) for which the mass is 25 grams.
2. Find the rate at which the mass is decreasing when \(t = 55\).

11 On the day that a new consumer product went on sale (day zero), a call centre received 1 call about it. On the 2nd day after day zero the call centre received 3 calls, and on the 10th day after day zero there were 200 calls. Two models were proposed to model \(N\), the number of calls received \(t\) days after day zero.
Model 1 is a linear model \(\mathrm { N } = \mathrm { mt } + \mathrm { c }\).

1. Determine the values of \(m\) and \(c\) which best model the data for 2 days and 10 days after day zero.
2. State the rate of increase in calls according to model 1.
3. Explain why this model is not suitable when \(t = 1\). Model 2 is an exponential model \(\mathbf { N } = e ^ { 0.53 t }\).
4. Verify that this is a good model for the number of calls when \(t = 2\) and \(t = 10\).
5. Determine the rate of increase in calls when \(t = 10\) according to model 2 .

3. Every \(\pounds 1\) of money invested in a savings scheme continuously gains interest at a rate of \(4 \%\) per year. Hence, after \(x\) years, the total value of an initial \(\pounds 1\) investment is \(\pounds y\), where $$y = 1.04 ^ { x }$$

1. Sketch the graph of \(y = 1.04 ^ { x } , x \geq 0\).
2. Calculate, to the nearest \(\pounds\), the total value of an initial \(\pounds 800\) investment after 10 years.
3. Use logarithms to find the number of years it takes to double the total value of any initial investment.

1 Solve the equation \(\frac { 3 ^ { x } + 2 } { 3 ^ { x } - 2 } = 8\), giving your answer correct to 3 decimal places.

15. The size \(N\) of the population of a small island at time \(t\) years may be modelled by \(N = A \mathrm { e } ^ { k t }\), where \(A\) and \(k\) are constants. It is known that \(N = 100\) when \(t = 2\) and that \(N = 160\) when \(t = 12\).

1. Interpret the constant \(A\) in the context of the question.
2. Show that \(k = 0 \cdot 047\), correct to three decimal places.
3. Find the size of the population when \(t = 20\).

1 Solve the equation \(4 \left| 5 ^ { x } - 1 \right| = 5 ^ { x }\), giving your answers correct to 3 decimal places.

1. The height, \(h\) metres, of a plant, \(t\) years after it was first measured, is modelled by the equation

$$h = 2.3 - 1.7 \mathrm { e } ^ { - 0.2 t } \quad t \in \mathbb { R } \quad t \geqslant 0$$ Using the model,

1. find the height of the plant when it was first measured,
2. show that, exactly 4 years after it was first measured, the plant was growing at approximately 15.3 cm per year. According to the model, there is a limit to the height to which this plant can grow.
3. Deduce the value of this limit.

4 The curve \(y = \mathrm { e } ^ { x } + 4 \mathrm { e } ^ { - 2 x }\) has one stationary point.

1. Find the \(x\)-coordinate of this point.
2. Determine whether the stationary point is a maximum or a minimum point.