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\$