could anyone give me a hand with these please
1 At the beginning of 1995, an investor decided to invest £6 000 in a plan that grew at an average rate of 6% per year. Show that, if this percentage rate of increase is maintained for 10 years, the value of the original investment will be about £10 745.
2 A population of bacteria is growing exponentially. Explain briefly what this means, using a sketch graph to illustrate your answer.
(i) At time t hours, the size, P, of the population of bacteria is given by P = 1000 (2.8)^0.02t. State the value of P when t = 0. Find the value of t when the size of the population is twice its initial value.
(ii) A second population of bacteria doubles in size every 12 hours. Find the number of hours it takes for this population to increase in size by a factor of 100.
At the beginning of 95,
Final Amount at the end of 95,
At the beginning of 96,
Final Amount at the end of 96,
At the beginning of 97,
Final Amount at the end of 97,
So, from here we can safely deduce that after n years, the amount will be
so after 10 years
This however is the long approach to doing things. If you can identify that it is a geometric progression from the question, you can straight use the formula
For the second set of problem
P = 1000(2.8)^0.02t
when t = 0
p = 1000(2.8)^0.02(0)
p = 1000
when p = 2000,
2000 = 1000(2.8)^0.02t
2 = 2.8^0.02t
ln 2 = 0.02t(ln 2.8)
t = ln 2/0.02(ln2.8)
let the initial population size = X
Population size = (X)(2^n), where n= number of times the bacteria multiplied
Given information, population size is p = 100X
100X = X(2^n)
n ln 2= ln 100
n = ln 100/ ln 2
n = 6.644 times
therefore time taken is 6.644 x 12 = 79.728 hours