The table below shows the number of molecules in experiment B.
Time(h) : 1st , 2nd , 3rd , 4th , 5th
No. of molecules : 3 , 8 , 15 , 24, 35
(i) how many molecules will there be in nth hour?
(ii) from which hour onwards would the no. of molecules be in experiment B?
I tried an exponential and that did not work so try a polynomial:
change in n per hour 5 7 9 11
change in change in n per hour = 2
constant second derivative --> parabola
n = a + k t + c t^2
dn/dt = k + 2 c t
d^2n/dt^2 = 2 c
so
2c = 2
c = 1
and
n = a + k t + t^2
when t = 1, n = 3
3 = a + k + 1
a + k = 2
when t = 2, n = 8
8 = a + 2 k + 4
a + 2 k = 4
subtract those 2 equations
-k = - 2
k = 2
so
n = a + 2 t + t^2
when t = 3, n = 15
15 = a + 2(3) + 9
a = 0
so
n = + 2 t + t^2
check when t = 5
n = 10 + 25 = 35 check
so I claim that
n = 2 t + t^2
To answer both questions, we need to identify the pattern or the relationship between the time (in hours) and the corresponding number of molecules in experiment B.
Looking at the table, we can observe that the number of molecules follows a specific pattern: the difference between consecutive numbers increases by 2 each time.
To find the number of molecules in the nth hour (question i), we can use this pattern:
- Subtract 1 from the hour number (n-1) to find the index of the hour in the sequence.
- The number of molecules can be calculated using the formula: (n-1)^2 + 2.
For example, to find the number of molecules in the 6th hour:
1. Subtract 1 from 6: 6 - 1 = 5. This tells us that the 6th hour is the 5th index in the sequence.
2. Plug this into the formula: (5)^2 + 2 = 25 + 2 = 27.
Therefore, there will be 27 molecules in the 6th hour.
To find from which hour onwards the number of molecules will be in experiment B (question ii), we need to identify the point where the pattern starts or the sequence begins.
In this case, by inspecting the given table, we can see that the sequence starts from the time (hour) 1. Therefore, the number of molecules will be in experiment B from the 1st hour onwards.
To summarize,
(i) The number of molecules in the nth hour can be calculated using the formula (n-1)^2 + 2.
(ii) The number of molecules will be in experiment B starting from the 1st hour onwards.