If Q represents the quantity of ice crystals measured over time, then calculate Q at t=1 seconds for k=-0.8 / second. Enter your answer in decimal notation (only). Round in the tenths place to give a fractional representation of a partly formed crystal (if any)

Q (1) = 103.25

Step-by-step explanation:

Given:

- The rate of ice crystals formation/depletion is given by:

Q' (t) = k * (Q - 70)

- For the initial conditions;

Q (0) = 144, k = - 0.8 / sec

Find:

Calculate Q at t=1 seconds for k=-0.8 / second. Enter your answer in decimal notation (only).

Solution:

- First we will solve the differential equation to get a functions Q with time, Q (t).

- Use the given ODE:

dQ / dt = - 0.8 * (Q - 70)

- Separate variables:

dQ / (Q - 70) = - 0.8*dt

- Integrate both sides:

Ln | Q - 70 | = - 0.8*t + C

- Use initial conditions to evaluate @ Q (0) = 144:

Ln | 144 - 70 | = - 0.8*0 + C

C = Ln | 74 |

- The Solution is given by:

Ln | (Q - 70) / 74 | = - 0.8*t

Q (t) - 70 = 74*e^ (-0.8*t)

Q (t) = 70 + 74*e^ (-0.8*t)

- Now use the solution to the ODE and evaluate @ t = 1s

Q (1) = 70 + 74*e^ (-0.8*1)

Q (1) = 103.25