Question

The rate of ice formation is (2- 0.3t) grams/minutes. If there was initial 10grams of ice the ice maker, find the mass of the ice after 10minutes

Answer 1

Since rate of ice formation depends on time, the simple equation

$$m={{m}_{in}}+v\cdot t$$

where m_in is the initial ice mass, v is the rate of ice formation, is wrong.

The mass of the ice after t minutes can be calculated by the equation

$$m={{m}_{in}}+\int\limits_{0}^{t}{v\left( \tau \right)d\tau }$$

We are given $v(t)=(2-0.3t)$grams/minutes. Thus

$$m={{m}_{in}}+\int\limits_{0}^{t}{\left( 2-0.3\tau \right)d\tau }$$

Integrate

$$m={{m}_{in}}+\left. \left( 2\tau -0.3\frac{{{\tau }^{2}}}{2} \right) \right|_{0}^{t} m={{m}_{in}}+2t-0.3\frac{{{t}^{2}}}{2}$$

Substitute m_in =10grams and t=10minutes

$$m=10+2\cdot 10-0.3\frac{{{10}^{2}}}{2}=10+20-15=15\,grams$$

Thus, the mass of ice after 10 minutes is equal to 15 grams