Tuesday, December 18, 2018

Application of exponent

$\def\D{\displaystyle}$

1 (CIE 2012, s, paper 12, question 9)
Variables $\D N$ and $\D x$ are such that $\D N = 200 + 50e^{\frac{x}{100}}.$
(i) Find the value of $\D N$ when $\D x = 0.$ [1]
(ii) Find the value of $\D x$ when $\D N = 600.$ [3]
(iii) Find the value of $\D N$ when $\D \frac{dN}{dx}=45.$ [4]

2 (CIE 2014, w, paper 21, question 5)
The number of bacteria $\D B$ in a culture, $\D t$ days after the first observation, is given by
\[B= 500 +400e^{0.2t}.\]
(i) Find the initial number present. [1]
(ii) Find the number present after 10 days. [1]
(iii) Find the rate at which the bacteria are increasing after 10 days. [2]
(iv) Find the value of $\D t$ when $\D B = 10000.$ [3]

3 (CIE 2014, w, paper 23, question 4)
The profit \$ $\D P$ made by a company in its nth year is modelled by \[P=1000e^{an+b}.\]
In the first year the company made \$2000 profit.
(i) Show that $\D a + b = \ln 2.$ [1]
In the second year the company made \$3297 profit.
(ii) Find another linear equation connecting $\D a$ and $\D b.$ [2]
(iii) Solve the two equations from parts (i) and (ii) to find the value of $\D a$ and of $\D b.$ [2]
(iv) Using your values for $\D a$ and $\D b,$ find the profit in the 10th year. [2]

4 (CIE 2016, w, paper 21, question 4)
The number of bacteria, $\D N,$ present in a culture can be modelled by the equation $\D N= 7000+ 2000e^{-0.05t},$ where $\D t$ is measured in days. Find
(i) the number of bacteria when $\D t = 10,$ [1]
(ii) the value of $\D t$ when the number of bacteria reaches 7500, [3]
(iii) the rate at which the number of bacteria is decreasing after 8 days. [3]

5 (CIE 2017, march, paper 12, question 11)
It is given that $\D y = Ae^{bx} ,$ where $\D A$ and $\D b$ are constants. When $\D \ln y$ is plotted against $\D x$ a straight line graph is obtained which passes through the points (1.0, 0.7) and (2.5, 3.7).
(i) Find the value of $\D A$ and of $\D b.$ [6]
(ii) Find the value of $\D y$ when $\D x = 2.$ [2]

6 (CIE 2017, march, paper 22, question 2)
The value, $\D V$ dollars, of a car aged $\D t$ years is given by $\D V = 12000 e^{-0.2t}.$
(i) Write down the value of the car when it was new. [1]
(ii) Find the time it takes for the value to decrease to $\D \frac{2}{3}$ of the value when it was new. [2]

7 (CIE 2017, s, paper 12, question 7)
It is given that $\D y = A(10^{bx}),$ where $\D A$ and $\D b$ are constants. The straight line graph obtained when $\D \lg y$ is plotted against $\D x$ passes through the points (0.5, 2.2) and (1.0, 3.7).
(i) Find the value of $\D A$ and of $\D b.$ [5]
Using your values of $\D A$ and $\D b,$ find
(ii) the value of $\D y$ when $\D x = 0.6,$ [2]
(iii) the value of $\D x$ when $\D y = 600.$ [2]

8 (CIE 2018, s, paper 11, question 5)
The population, $\D P,$ of a certain bacterium $\D t$ days after the start of an experiment is modelled by $\D P = 800e^{kt},$ where $\D k$ is a constant.
(i) State what the figure 800 represents in this experiment. [1]
(ii) Given that the population is 20 000 two days after the start of the experiment, calculate the value of $\D k.$ [3]
(iii) Calculate the population three days after the start of the experiment. [2]

9 (CIE 2018, s, paper 12, question 7)
A population, $\D B,$ of a particular bacterium, $\D t$ hours after measurements began, is given by $\D B =1000e^{\frac{t}{4}}.$
(i) Find the value of $\D B$ when $\D t = 0.$ [1]
(ii) Find the time taken for $\D B$ to double in size. [3]
(iii) Find the value of $\D B$ when $\D t = 8.$ [1]


1. $\D 250;208;4700$
2. (i) $\D 900$
(ii) $\D 3456$
(iii) $\D 591$
(iv) $\D 15.8$
3. (ii) $\D 2a + b = \ln 3.297$
(iii) $\D a = 0.5; b = 0.193$
(iv) $\D n = 10; P = 180000$
4. (i) $\D 8213$
(ii) $\D 27.7$
(iii) $\D \pm 67$
5. (i) $\D b = 2;A = 0.273,$
(ii) $\D 14.9$
6. (i) $\D 12 000$
(ii) $\D 2$
7. (i) $\D b=3,A=5.01$ or $\D 10.7$
(ii) $\D y=315$ or $\D 102.5$
(iii) $\D x = 0.693$
8. The number of bacteria at the
start of the experiment
$\D 1.61,100 000$
9. (i) $\D 1000,$
(ii) $\D 2.77$
(iii) $\D 7389$

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