A cafe roasts its own coffee beans and packages them to be sold. The amount of roasted coffee beans

Question

A cafe roasts its own coffee beans and packages them to be sold. The amount of roasted coffee beans remaining in the cafe at time $t$ days is denoted by $x$ kg. The cafe produces roasted coffee beans at a fixed rate of $10$ kg/day, and sells the roasted coffee beans at a rate proportional to $x^2$ kg/day. Given that there is no change in the amount of roasted coffee beans remaining in the cafe when there is $5$ kg of beans remaining, show that $\frac{\mathrm{d}x}{\mathrm{d}t} = 10 - \frac{2}{5}x^2$. Find the general solution of this differential equation. (You do not have to make $x$ the subject.)[3] Due to improvement in roasting capabilities of the cafe, it is now able to produce roasted coffee beans at the rate of $3x$ kg/day, and sells the roasted coffee beans at a rate proportional to $x^2$ kg/day. Given that amount of roasted coffee beans is increasing at the rate of $6$ kg/day when the amount of roasted coffee beans remaining in the cafe is $6$ kg, write down the modified differential equation relating $\frac{\mathrm{d}x}{\mathrm{d}t}$ and $x$.[2] Given that the initial amount of roasted coffee beans in the cafe is $13.5$ kg, solve the differential equation in part (b), expressing $x$ in terms of $t$. Deduce the amount of roasted coffee beans remaining in the cafe in the long run, and sketch the curve of $x$ against $t$.[6] Based on the curve in part (c), explain in context why this might be favourable for the cafe owner.[1]

Accepted Answer

(a)$\frac{1}{2(5)}\ln |\frac{x-5}{x+5}|=-\frac{2}{5}t+c$ (b)$\frac{\mathrm{d}x}{\mathrm{d}t}=3x-\frac{1}{3}x^2$ (c)$x=\frac{27e^{3t}}{3e^{3t}-1}$

2025 RVHS P1 Q13

2025 RVHS P1 Q13