# A level differentiation question

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#1
Hey, I'm struggling with the second part of this exam question and would appreciate any guidance:

The number of mice, N, in a population t months after the start of a study is modelled by the equation:
N = 900 / (3 + 7e^-0.25t) [t is real, t is greater than/equal to 0]

(a) Show that the rate of growth dN/dt is given by
dN/dt = N(300-N) / 1200
(which I have done) - then the next part I don't understand:

(b) The rate of growth is a maximum after T months. Find, according to the model, the value of T.

The mark scheme says that the first step is 'Deduces or shows that dN/dt is maximised when N = 150' - I get how you work it out from there but I don't understand where the 150 comes from?

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2 years ago
#2
(Original post by e123c)
Hey, I'm struggling with the second part of this exam question and would appreciate any guidance:

The number of mice, N, in a population t months after the start of a study is modelled by the equation:
N = 900 / (3 + 7e^-0.25t) [t is real, t is greater than/equal to 0]

(a) Show that the rate of growth dN/dt is given by
dN/dt = N(300-N) / 1200
(which I have done) - then the next part I don't understand:

(b) The rate of growth is a maximum after T months. Find, according to the model, the value of T.

The mark scheme says that the first step is 'Deduces or shows that dN/dt is maximised when N = 150' - I get how you work it out from there but I don't understand where the 150 comes from?

You're looking for when dN/dt is a maximum - note: not when N is a maximum.

Couple of ways you can go:

1. You could recognize that the denominator is a constant, and the numerator is a quadratic. So, it's a maximum when the numerator is a maximum.
The quadratic has roots 0 and 300 (if you set it equal to 0). The max (or min) of a quadratic occurs at the midpoint of its two roots, i.e. (0+300)/2=150.

2. Differentiate again (with respect to N this time), set equal to 0, to find the stationary point of dN/dt, and that's where the max/min for dN/dt is.
Last edited by ghostwalker; 2 years ago
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#3
Ohh okay that makes sense Thank you
(Original post by ghostwalker)
You're looking for when dN/dt is a maximum - note: not when N is a maximum.

Couple of ways you can go:

1. You could recognize that the denominator is a constant, and the numerator is a quadratic. So, it's a maximum when the numerator is a maximum.
The quadratic has roots 0 and 300 (if you set it equal to 0). The max (or min) of a quadratic occurs at the midpoint of its two roots, i.e. (0+300)/2=150.

2. Differentiate again (with respect to N this time), set equal to 0, to find the stationary point of dN/dt, and that's where the max/min for dN/dt is.
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