(a) The texture of a food changes during cooking. This process is modelled as an

Question

(a) The texture of a food changes during cooking. This process is modelled as an initial
value problem:
(dP/dt)= -kP P(0) = P0
where P is a measure of the texture and t is the cooking time elapsed (in minutes). P
is chosen to have units in which P0 = 1. The rate parameter k is assumed to have the
constant value k = 0:03.
(i) Use Euler's method with a single step (of ten minutes) to estimate the value of the
texture measure P after ten minutes cooking.
(ii) Find an explicit mathematical expression for P (t). Show your working.
(iii) Use your explicit expression for P (t) to find the value of P after ten minutes cooking.
(iv) How long will it take for P to decrease to half its original value.

Explanation / Answer

i)P(10)= 1 + -0.03*10= 1-.3= 0.7 ii) dP/P= -0.03 dt ln P= -0.03 t +C e^(ln P) = e^(-.03 t+C) P= Ce^(-0.03t) Since P(0)= 1,P= e^(-0.03t). iii) P= e^(-.3)= 0.7408 iv) .5 = e^(-.03t) t= ln .5/-.03= 23.1 minutes

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