1.06g Equations with exponentials: solve a^x = b

4. The growth of a weed on the surface of a pond is being studied. The surface area of the pond covered by the weed, \(A \mathrm {~m} ^ { 2 }\), is modelled by the equation $$A = \frac { 80 p \mathrm { e } ^ { 0.15 t } } { p \mathrm { e } ^ { 0.15 t } + 4 }$$ where \(p\) is a positive constant and \(t\) is the number of days after the start of the study.
Given that

• \(30 \mathrm {~m} ^ { 2 }\) of the surface of the pond was covered by the weed at the start of the study
• \(50 \mathrm {~m} ^ { 2 }\) of the surface of the pond was covered by the weed \(T\) days after the start of the study
  1. show that \(p = 2.4\)
  2. find the value of \(T\), giving your answer to one decimal place.
     (Solutions relying entirely on graphical or numerical methods are not acceptable.)

The weed grows until it covers the surface of the pond.

Find, according to the model, the maximum possible surface area of the pond.

8. A dose of antibiotics is given to a patient. The amount of the antibiotic, \(x\) milligrams, in the patient's bloodstream \(t\) hours after the dose was given, is found to satisfy the equation $$\log _ { 10 } x = 2.74 - 0.079 t$$

1. Show that this equation can be written in the form $$x = p q ^ { - t }$$ where \(p\) and \(q\) are constants to be found. Give the value of \(p\) to the nearest whole number and the value of \(q\) to 2 significant figures.
2. With reference to the equation in part (a), interpret the value of the constant \(p\). When a different dose of the antibiotic is given to another patient, the values of \(x\) and \(t\) satisfy the equation $$x = 400 \times 1.4 ^ { - t }$$
3. Use calculus to find, to 2 significant figures, the value of \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) when \(t = 5\)

3. \begin{figure}[h] \end{figure} The line \(l\) in Figure 1 shows a linear relationship between \(\log _ { 10 } y\) and \(x\).
The line passes through the points \(( 0,1.5 )\) and \(( - 4.8,0 )\) as shown.

1. Write down an equation for \(l\).
2. Hence, or otherwise, express \(y\) in the form \(k b ^ { x }\), giving the values of the constants \(k\) and \(b\) to 3 significant figures.

1. The amount of money raised for a charity is being monitored.

The total amount raised in the \(t\) months after monitoring began, \(\pounds D\), is modelled by the equation $$\log _ { 10 } D = 1.04 + 0.38 t$$

1. Write this equation in the form $$D = a b ^ { t }$$ where \(a\) and \(b\) are constants to be found. Give each value to 4 significant figures. When \(t = T\), the total amount of money raised is \(\pounds 45000\) According to the model,
2. find the value of \(T\), giving your answer to 3 significant figures. The charity aims to raise a total of \(\pounds 350000\) within the first 12 months of monitoring.
   According to the model,
3. determine whether or not the charity will achieve its aim.

10. The amount of an antibiotic in the bloodstream, from a given dose, is modelled by the formula $$x = D \mathrm { e } ^ { - 0.2 t }$$ where \(x\) is the amount of the antibiotic in the bloodstream in milligrams, \(D\) is the dose given in milligrams and \(t\) is the time in hours after the antibiotic has been given. A first dose of 15 mg of the antibiotic is given.

1. Use the model to find the amount of the antibiotic in the bloodstream 4 hours after the dose is given. Give your answer in mg to 3 decimal places. A second dose of 15 mg is given 5 hours after the first dose has been given. Using the same model for the second dose,
2. show that the total amount of the antibiotic in the bloodstream 2 hours after the second dose is given is 13.754 mg to 3 decimal places. No more doses of the antibiotic are given. At time \(T\) hours after the second dose is given, the total amount of the antibiotic in the bloodstream is 7.5 mg .
3. Show that \(T = a \ln \left( b + \frac { b } { \mathrm { e } } \right)\), where \(a\) and \(b\) are integers to be determined.
   

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 \(V = 15000\)

6. A hot piece of metal is dropped into a cool liquid. As the metal cools, its temperature \(T\) degrees Celsius, \(t\) minutes after it enters the liquid, is modelled by $$T = 300 \mathrm { e } ^ { - 0.04 t } + 20 , \quad t \geqslant 0$$

1. Find the temperature of the piece of metal as it enters the liquid.
2. Find the value of \(t\) for which \(T = 180\), giving your answer to 3 significant figures. (Solutions based entirely on graphical or numerical methods are not acceptable.)
3. Show, by differentiation, that the rate, in degrees Celsius per minute, at which the temperature of the metal is changing, is given by the expression $$\frac { 20 - T } { 25 }$$

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10. A population of insects is being studied. The number of insects, \(N\), in the population, is modelled by the equation $$N = \frac { 300 } { 3 + 17 \mathrm { e } ^ { - 0.2 t } } \quad t \in \mathbb { R } , t \geqslant 0$$ where \(t\) is the time, in weeks, from the start of the study.
Using the model,

1. find the number of insects at the start of the study,
2. find the number of insects when \(t = 10\),
3. find the time from the start of the study when there are 82 insects. (Solutions based entirely on graphical or numerical methods are not acceptable.)
4. Find, by differentiating, the rate, measured in insects per week, at which the number of insects is increasing when \(t = 5\). Give your answer to the nearest whole number.

6. The mass, \(m\) grams, of a radioactive substance \(t\) years after first being observed, is modelled by the equation $$m = 25 \mathrm { e } ^ { 1 - k t }$$ where \(k\) is a positive constant.

1. State the value of \(m\) when the radioactive substance was first observed. Given that the mass is 50 grams, 10 years after first being observed,
2. show that \(k = \frac { 1 } { 10 } \ln \left( \frac { 1 } { 2 } \mathrm { e } \right)\)
3. Find the value of \(t\) when \(m = 20\), giving your answer to the nearest year.

3. $$f ( x ) = 2 ^ { x - 1 } - 4 + 1.5 x \quad x \in \mathbb { R }$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as $$x = \frac { 1 } { 3 } \left( 8 - 2 ^ { x } \right)$$ The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\), where \(\alpha = 1.6\) to one decimal place.
2. Starting with \(x _ { 0 } = 1.6\), use the iteration formula $$x _ { n + 1 } = \frac { 1 } { 3 } \left( 8 - 2 ^ { x _ { n } } \right)$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
3. By choosing a suitable interval, prove that \(\alpha = 1.633\) to 3 decimal places.

9. A rare species of mammal is being studied. The population \(P\), \(t\) years after the study started, is modelled by the formula $$P = \frac { 900 \mathrm { e } ^ { \frac { 1 } { 4 } t } } { 3 \mathrm { e } ^ { \frac { 1 } { 4 } t } - 1 } , \quad t \in \mathbb { R } , \quad t \geqslant 0$$ Using the model,

1. calculate the number of mammals at the start of the study,
2. calculate the exact value of \(t\) when \(P = 315\) Give your answer in the form \(a \ln k\), where \(a\) and \(k\) are integers to be determined.
3. 1. Find \(\frac { \mathrm { d } P } { \mathrm {~d} t }\)
   2. Hence find the value of \(\frac { \mathrm { d } P } { \mathrm {~d} t }\) when \(t = 8\), giving your answer to 2 decimal places.

9. (i) Find the exact solutions to the equations

1. \(\ln ( 3 x - 7 ) = 5\)
2. \(3 ^ { x } \mathrm { e } ^ { 7 x + 2 } = 15\) (ii) The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } + 3 , & x \in \mathbb { R } \\ \mathrm {~g} ( x ) = \ln ( x - 1 ) , & x \in \mathbb { R } , x > 1 \end{array}$$
   1. Find \(\mathrm { f } ^ { - 1 }\) and state its domain.
   2. Find fg and state its range.

4. Joan brings a cup of hot tea into a room and places the cup on a table. At time \(t\) minutes after Joan places the cup on the table, the temperature, \(\theta ^ { \circ } \mathrm { C }\), of the tea is modelled by the equation $$\theta = 20 + A \mathrm { e } ^ { - k t } ,$$ where \(A\) and \(k\) are positive constants. Given that the initial temperature of the tea was \(90 ^ { \circ } \mathrm { C }\),

1. find the value of \(A\). The tea takes 5 minutes to decrease in temperature from \(90 ^ { \circ } \mathrm { C }\) to \(55 ^ { \circ } \mathrm { C }\).
2. Show that \(k = \frac { 1 } { 5 } \ln 2\).
3. Find the rate at which the temperature of the tea is decreasing at the instant when \(t = 10\). Give your answer, in \({ } ^ { \circ } \mathrm { C }\) per minute, to 3 decimal places.

3. The area, \(A \mathrm {~mm} ^ { 2 }\), of a bacterial culture growing in milk, \(t\) hours after midday, is given by $$A = 20 \mathrm { e } ^ { 1.5 t } , \quad t \geqslant 0$$

1. Write down the area of the culture at midday.
2. Find the time at which the area of the culture is twice its area at midday. Give your answer to the nearest minute.

1. The function f is defined by

$$\mathrm { f } : x \mapsto \frac { 3 ( x + 1 ) } { 2 x ^ { 2 } + 7 x - 4 } - \frac { 1 } { x + 4 } , \quad x \in \mathbb { R } , x > \frac { 1 } { 2 }$$

1. Show that \(\mathrm { f } ( x ) = \frac { 1 } { 2 x - 1 }\)
2. Find \(\mathrm { f } ^ { - 1 } ( x )\)
3. Find the domain of \(\mathrm { f } ^ { - 1 }\) $$\mathrm { g } ( x ) = \ln ( x + 1 )$$
4. Find the solution of \(\mathrm { fg } ( x ) = \frac { 1 } { 7 }\), giving your answer in terms of e .

2. $$\mathrm { g } ( x ) = \mathrm { e } ^ { x - 1 } + x - 6$$

1. Show that the equation \(\mathrm { g } ( x ) = 0\) can be written as $$x = \ln ( 6 - x ) + 1 , \quad x < 6$$ The root of \(\mathrm { g } ( x ) = 0\) is \(\alpha\).
   The iterative formula $$x _ { n + 1 } = \ln \left( 6 - x _ { n } \right) + 1 , \quad x _ { 0 } = 2$$ is used to find an approximate value for \(\alpha\).
2. Calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) to 4 decimal places.
3. By choosing a suitable interval, show that \(\alpha = 2.307\) correct to 3 decimal places.

8. The function \(f\) is defined by $$\mathrm { f } : x \rightarrow 3 - 2 \mathrm { e } ^ { - x } , \quad x \in \mathbb { R }$$

1. Find the inverse function, \(\mathrm { f } ^ { - 1 } ( x )\) and give its domain.
2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = \ln x\). The equation \(\mathrm { f } ( t ) = k \mathrm { e } ^ { t }\), where \(k\) is a positive constant, has exactly one real solution.
3. Find the value of \(k\).

1. A particular species of orchid is being studied. The population \(p\) at time \(t\) years after the study started is assumed to be

$$p = \frac { 2800 a \mathrm { e } ^ { 0.2 t } } { 1 + a \mathrm { e } ^ { 0.2 t } } , \text { where } a \text { is a constant. }$$ Given that there were 300 orchids when the study started,

1. show that \(a = 0.12\),
2. use the equation with \(a = 0.12\) to predict the number of years before the population of orchids reaches 1850.
3. Show that \(p = \frac { 336 } { 0.12 + \mathrm { e } ^ { - 0.2 t } }\).
4. Hence show that the population cannot exceed 2800.
   

Find the exact solutions to the equations

1. \(\ln x + \ln 3 = \ln 6\),
2. \(\mathrm { e } ^ { x } + 3 \mathrm { e } ^ { - x } = 4\).

6. Find algebraically the exact solutions to the equations

1. \(\ln ( 4 - 2 x ) + \ln ( 9 - 3 x ) = 2 \ln ( x + 1 ) , \quad - 1 < x < 2\)
2. \(2 ^ { x } \mathrm { e } ^ { 3 x + 1 } = 10\) Give your answer to (b) in the form \(\frac { a + \ln b } { c + \ln d }\) where \(a , b , c\) and \(d\) are integers.

6. The function f is defined by $$\mathrm { f } : x \rightarrow \mathrm { e } ^ { 2 x } + k ^ { 2 } , \quad x \in \mathbb { R } , \quad k \text { is a positive constant. }$$

1. State the range of f .
2. Find \(\mathrm { f } ^ { - 1 }\) and state its domain. The function g is defined by $$g : x \rightarrow \ln ( 2 x ) , \quad x > 0$$
3. Solve the equation $$\mathrm { g } ( x ) + \mathrm { g } \left( x ^ { 2 } \right) + \mathrm { g } \left( x ^ { 3 } \right) = 6$$ giving your answer in its simplest form.
4. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
5. Find, in terms of the constant \(k\), the solution of the equation $$\mathrm { fg } ( x ) = 2 k ^ { 2 }$$

2. Find the exact solutions, in their simplest form, to the equations

1. \(2 \ln ( 2 x + 1 ) - 10 = 0\)
2. \(3 ^ { x } \mathrm { e } ^ { 4 x } = \mathrm { e } ^ { 7 }\)

8. A rare species of primrose is being studied. The population, \(P\), of primroses at time \(t\) years after the study started is modelled by the equation $$P = \frac { 800 \mathrm { e } ^ { 0.1 t } } { 1 + 3 \mathrm { e } ^ { 0.1 t } } , \quad t \geqslant 0 , \quad t \in \mathbb { R }$$

1. Calculate the number of primroses at the start of the study.
2. Find the exact value of \(t\) when \(P = 250\), giving your answer in the form \(a \ln ( b )\) where \(a\) and \(b\) are integers.
3. Find the exact value of \(\frac { \mathrm { d } P } { \mathrm {~d} t }\) when \(t = 10\). Give your answer in its simplest form.
4. Explain why the population of primroses can never be 270

2. Given that $$\mathrm { f } ( x ) = 2 \mathrm { e } ^ { x } - 5 , \quad x \in \mathbb { R }$$

1. sketch, on separate diagrams, the curve with equation
   1. \(y = \mathrm { f } ( x )\)
   2. \(y = | \mathrm { f } ( x ) |\) On each diagram, show the coordinates of each point at which the curve meets or cuts the axes. On each diagram state the equation of the asymptote.
2. Deduce the set of values of \(x\) for which \(\mathrm { f } ( x ) = | \mathrm { f } ( x ) |\)
3. Find the exact solutions of the equation \(| \mathrm { f } ( x ) | = 2\)

1. Water is being heated in an electric kettle. The temperature, \(\theta ^ { \circ } \mathrm { C }\), of the water \(t\) seconds after the kettle is switched on, is modelled by the equation

$$\theta = 120 - 100 \mathrm { e } ^ { - \lambda t } , \quad 0 \leqslant t \leqslant T$$

1. State the value of \(\theta\) when \(t = 0\) Given that the temperature of the water in the kettle is \(70 ^ { \circ } \mathrm { C }\) when \(t = 40\),
2. find the exact value of \(\lambda\), giving your answer in the form \(\frac { \ln a } { b }\), where \(a\) and \(b\) are integers. When \(t = T\), the temperature of the water reaches \(100 ^ { \circ } \mathrm { C }\) and the kettle switches off.
3. Calculate the value of \(T\) to the nearest whole number.