estimate the amount of solid

Question

Let f(t) be the weight (in grams) of a solid sitting in a beaker of water. Suppose that the solid dissolves in such a way that the rate of change (in grams/minute) of the weight of the solid at any time t can be determined from the weight using the formula: f'(t)=-1f(t)(3+f(t)) If there is 2 grams of solid at time t=2 estimate the amount of solid 1 second later.

Answer 1

f'(t) = –f(t)(3+f(t)) g/min and

there are 2 g of solid left after 2 min, so f(2) = 2 g.

This means that the rate of change at t = 2 is

f'(2) = –f(2)(3+f(2)) = –2(3+2) = –10 g/min

To determine the amount of solid 1 second later we

first divide by 60 to determine the rate in g/s

(–10 g/min)/(60 s/min) = –1/6 g/s

So at t = 2 min, we can estimate that 1/6 g of solid will dissolve in the next second.

This means that the amount of solid remaining will be

2 – 1/6 = 12/6 – 1/6 = 11/6 g.

Answer

At time t=2 we can estimate that 1 second later the amount of solid will be 11/6 ≈ 1.83 g