Question

A calf that weighs 60 pounds at birth gains weight at the rate dw/dt = k(1200 - w)

Answer 1

Dw/dt = k(1200 - w)
dw/(1200 - w) = kdt
-ln(1200 - w) = kt + c

When t = 0, w = 60
-ln(1200 - 60) = c
c = -ln(1140)

a.) For k = 0.8: -ln(1200 - w) = 0.8t - ln(1140)
t =  [ln(1140) - ln(1200 - w)]/0.8

For k = 0.9:
t = [ln(1140) - ln(1200 - w)]/0.9

For k = 1:
t = [ln(1140) - ln(1200 - w)]
 
b.) For k = 0.8
t = [ln(1140) - ln(1200 - 800)]/0.8 = [ln(1140) - ln(400)]/0.8 = 1.3 years

For k = 0.9
t = [ln(1140) - ln(400)]/0.9 = 1.16 years

For k = 1
t = [ln(1140) - ln(400)] = 1.05 years

c.) For maximum weight,
dw/dt = 0
k(1200 - w) = 0
1200 - w = 0
w = 1200

Therefore, the maximum weight for each of the model is 1200 pounds.