Question

A yeast culture weighing 2 grams is removed from a refrigerator unit and is expected to grow at the rate of Upper W prime (t )equals0.1 e Superscript 0.1 t grams per hour at a higher controlled temperature. How much will the weight of the culture increase during the first 10 hours of​ growth? How much will the weight of the culture increase from the end of the 10th hour to the end of the 20th hour of​ growth?

Answer 1

a) during the first 10 hours of growth the weight will increase 271.8% relative to the initial state ( or 2.718 g)

b) during the first 20 hours of growth the weight will increase 467.1 % relative to the initial state ( or 4.671 g)

Explanation:

since the growing rate law is

W'(t) = 0.1 gr/hour * e^(0.1gr/hour* t) , W'(t) [gr/hour]

and following mathematical conventions: W'(t)= dW/dt

then

dW/dt=0.1 e^(0.1t)

∫dW =∫0.1 e^(0.1t) dt

W = e^(0.1t) + C

at the beginning, (time t=0) the weight is W=2 grams .Therefore

2 g = e^(0.1 g/h*0) + C → 2 g = 1 g + C → C = 1 g

then

W = e^(0.1t) + 1 g

at t= 10 hours

W = e^(0.1 g/h*10h) + 1 g = 3.718 g/h

therefore the weight will increase

ΔW = 3.718 g - 1 g = 2.718 g or 271.8% relative to the initial state

for t=20 hours

W = e^(0.1 g/h*20h) + 1 g = 8.389 g/h

thus, the from t= 10 hours to t= 20 hours the weight will increase

ΔW = 8.389 g/h - 3.718 g = 4.671 g or 467.1 %relative to the initial state