미분방정식 전체 문제

• First scenario: At time $0$, Newton mixes 9 parts coffee at temperature $95^\circ$ C with 1 part cream at temperature $5^\circ$ C. The resulting mixture has temperature $$\frac{9\times 95+1\times 5}{9+1}=86^\circ$$ The mixture cools according to Newton's Law of Cooling, with initial temperature 86$^\circ$ and ambient temperature 22$^\circ$:

$$\begin{aligned} T(t)&=[86-22]e^{-kt}+22\\ T(t)&=64e^{-kt}+22 \\ \text{After 10 minutes,} \\ 54=T(10)&=22+64e^{-10 k}\\ e^{-10 k}&=\frac{54-22}{64}=\frac{1}{2} \end{aligned}$$

We could compute $k$ from this, but we don't need it.

• Second scenario: At time $0$, Newton gets hot coffee at temperature $95^\circ$ C. It cools according to Newton's Law of Cooling

$$\begin{aligned} T(t)=[T(0)-22]e^{-kt}+22 \end{aligned}$$

In this second scenario, $T(0)=95$, so $$T(t)=[95-22]e^{-kt}+22=73e^{-kt}+22$$ The value of $k$ is the same as in the first scenario, so after 10 minutes $$T(10)=22+73e^{-10k}=22+73\frac{1}{2}=58.5$$ This cooled coffee is mixed with cold cream to yield a mixture of temperature $$\frac{9\times 58.5+1\times 5}{9+1}=53.15$$

Under the second (add cream just before drinking) scenario, the coffee ends up cooler by $0.85^\circ$ C$\,$.

(a) By Newton's law of cooling, the rate of change of temperature is proportional to the difference between $T(t)$ and the ambient temperature, which in this case is $30^\circ$. Thus $$\frac{\mathrm{d}T}{\mathrm{d}t}=k[T(t)-30]$$ for some constant of proportionality $k$. To answer part (a), all we have to do is find $k$.

Under Newton's Law of Cooling, the temperature at time $t$ will be given by

$$\begin{aligned} T(t)&=[T(0)-A]e^{kt}+A\\ &=[5-30]e^{kt}+30\\ &=-25e^{kt}+30 \\ \text{We are told $T(5)=10$:} \\ 10&=-25e^{5k}+30\\ 25e^{5k}&=20\\ e^{5k}&=\frac{4}{5}\\ 5k&=\log(4/5)\\ k&=\tfrac{1}{5}\log(4/5) \\ \text{So, the differential equation is} \\ \frac{\mathrm{d}T}{\mathrm{d}t}(t)&=\frac{1}{5}\log(4/5)[T(t)-30] \end{aligned}$$ (b) Since $T(t)=30-25e^{kt}$, the temperature of the tea is $20^\circ$ when

$$\begin{aligned} 30-25 e^{kt}&=20\\ e^{kt}&=\frac{10}{25}\\ kt&=\log\left(\frac{10}{25}\right)\\ t&=\frac{1}{k}\log\frac{2}{5}\\ &=\frac{5\log(2/5)}{\log(4/5)}\\ &\approx{20.53 ~\mathrm{ min}} \end{aligned}$$

Suppose that in a very short time interval $\mathrm{d}t$, the height of water in the tank changes by $\mathrm{d}h$ (which is negative). Then in this time interval the amount of the water in the tank decreases by $\mathrm{d}V=-\pi(3)^2\mathrm{d}h$. This must be the same as the amount of water that flows through the hole in this time interval. The water flowing through the hole makes a cylinder of radius 1 cm (that is, 0.01 m) with length $v(t)\mathrm{d}t$, the distance the water moves out of the hole in $\mathrm{d}t$ seconds. So, the amount of water leaving the hole over the time interval $\mathrm{d}t$ is $\pi(0.01)^2 v(t)\,\mathrm{d}t =\pi(0.01)^2 \sqrt{2gh(t)}\,\mathrm{d}t$.

This gives us a separable differential equation. Recall $g$ is a constant.

$$\begin{aligned} -\pi(3)^2\mathrm{d}h&=\pi(0.01)^2 \sqrt{2gh(t)}\,\mathrm{d}t\\ \frac{\mathrm{d}h}{\sqrt{h}}&=-{\Big(\frac{0.01}{3}\Big)}^{\!2} \sqrt{2g}\,\mathrm{d}t\\ \int\frac{\mathrm{d}h}{\sqrt{h}}&=\int-{\Big(\frac{0.01}{3}\Big)}^{\!2} \sqrt{2g}\,\mathrm{d}t\\ 2\sqrt{h}&=-{\Big(\frac{0.01}{3}\Big)}^{\!2} \sqrt{2g}\,t+C \\ \text{At time $0$, the height is $6$, so $C=2\sqrt{6}$ and} \\ 2\sqrt{h}&=-{\Big(\frac{0.01}{3}\Big)}^{\!2} \sqrt{2g}\,t+2\sqrt{6} \\ \text{We want to know when the height of the water in the tank is 0.} \\ 0&=-\Big(\frac{0.01}{3}\Big)^{\!2} \sqrt{2g}\,t+2\sqrt{6}\\ \Big(\frac{0.01}{3}\Big)^{\!2} \sqrt{2g}\,t&=2\sqrt{6}\\ t&=\frac{2\sqrt6}{\Big(\frac{0.01}{3}\Big)^{\!2} \sqrt{2g}}\\ &=2{\Big(\frac{3}{0.01}\Big)}^{\!2} \sqrt{\frac{3}{g}}\\ &=180,000 \sqrt{\frac{3}{g}}\approx 99,591\text{ sec} \approx 27.66\text{ hr} \end{aligned}$$

Suppose that at time $t$, the mercury in the tank has height $h$, which is between 0 and 12 feet.

At that time, the top surface of the mercury forms a circular disk of radius $\sqrt{6^2-(h-6)^2}$. (We found this by applying the Pythagorean Theorem to the triangle in the diagram above. In the diagram, $h$ is shown as being larger than 6, but the same equation holds for all $h$ in $[0,12]$.) Now suppose that in a very short time interval $\mathrm{d}t$, the height of mercury in the tank changes by $\mathrm{d}h$ (which is negative). Then in this time interval the amount of the mercury in the tank decreases by $-\pi\big(\sqrt{6^2-(h-6)^2}\ \big)^2\mathrm{d}h$. (That's the volume of the red disk in the figure above.) This must be the same as the amount of mercury that flows through the hole in this time interval. The mercury comes out of the hole as a cylinder. Its radius is the radius of the hole, $\frac{1}{12}$ foot, and its length is the distance the mercury travels in $\mathrm{d}t$ seconds, $v(t)\mathrm{d}t$ feet. So, the volume of escaped mercury is $\pi\big(\frac{1}{12}\big)^2 v\,\mathrm{d}t =\pi\big(\frac{1}{12}\big)^2 \sqrt{2gh}\,\mathrm{d}t$. This gives us a separable differential equation.

$$\begin{aligned} -\pi{\big(\sqrt{6^2-(h-6)^2}\ \big)}^2\mathrm{d}h &=\pi\Big(\frac{1}{12}\Big)^2 \sqrt{2gh}\,\mathrm{d}t\\ -{\big({36-(h^2-12h+36)}\ \big)}\mathrm{d}h &=\Big(\frac{1}{12}\Big)^2 \sqrt{2gh}\,\mathrm{d}t\\ \big(h^2-12h\big)\mathrm{d}h &=\frac{1}{144} \sqrt{2g}\sqrt{h}\,\mathrm{d}t\\ \big(h^{3/2}-12h^{1/2}\big)\mathrm{d}h&=\frac{1}{144} \sqrt{2g}\,\mathrm{d}t\\ \int \big(h^{3/2}-12h^{1/2}\big)\mathrm{d}h&=\int\frac{1}{144} \sqrt{2g}\,\mathrm{d}t\\ \frac{h^{5/2}}{5/2}-12\frac{h^{3/2}}{3/2}&=\frac{1}{144} \sqrt{2g}\,t+C \\ \text{At time $0$, the height is $12$, so $C=\frac{12^{5/2}}{5/2}-12\frac{12^{3/2}}{3/2} =12^{5/2}\big(\frac{2}{5}-\frac{2}{3}\big) =-\frac{4}{15}12^{5/2}$, which yields} \\ \frac{h^{5/2}}{5/2}-12\frac{h^{3/2}}{3/2}&=\frac{1}{144} \sqrt{2g}\,t -\frac{4}{15}12^{5/2} \\ \text{We want to find the time $t$ when the height is $h=0$.} \\ 0&=\frac{1}{144} \sqrt{2g}\,t-\frac{4}{15}12^{5/2}\\ \frac{1}{144} \sqrt{2g}\,t&=\frac{4}{15}12^{5/2}\\ t&=\frac{4\times 144}{15} \sqrt{\frac{12^5}{2g}}\\ &=38.4 \sqrt{\frac{124416}{g}}\approx 2,394\,\text{sec }\approx 0.665\, \text{hr} \end{aligned}$$

Suppose that at time $t$ (measured in hours starting at, say, noon), the water in the tank has height $y$, which is between 0 and 2 metres. At that time, the top surface of the water forms a circular disk of radius $r=y^p$ and area $A(y)=\pi y^{2p}$. Thus, by Torricelli's law,

$$\begin{aligned} \pi y^{2p}\frac{\mathrm{d}y}{\mathrm{d}t}&=-c\sqrt{y}\\ -\frac{\pi}{c}\cdot y^{2p-{1\over 2}}\mathrm{d}y&=\,\mathrm{d}t\\ \int-\frac{\pi}{c}\cdot y^{2p-{1\over 2}}\mathrm{d}y&=\,\int \mathrm{d}t\\ -\frac{\pi}{c}\cdot\frac{y^{2p+{1\over 2}}}{2p+{1\over 2}}+d &=t \\ \text{for some constant $d$. At time $t=0$, the height is $y=2$, so $d=\displaystyle\frac{\pi}{c}\cdot\frac{2^{2p+{1\over 2}}}{2p+{1\over 2}}$ \ .} \\ t&=\frac{\pi}{c}\bigg(\frac{2^{2p+{1\over 2}}}{2p+{1\over 2}} -\frac{y^{2p+{1\over 2}}}{2p+{1\over 2}}\bigg)\\ &=\frac{\pi}{c(2p+\frac12)}\left(2^{2p+\frac12}-y^{2p+\frac12}\right) \end{aligned}$$

The time at which the height is $1$ is obtained by subbing $y=1$ into this formula. The time at which the height is $0$ is obtained by subbing $y=0$ into this formula. Thus the condition that the top half ($y=2$ to $y=1$) takes exactly the same amount of time to drain as the bottom half ($y=1$ to $y=0$) is:

$$\begin{aligned} t(2)-t(1)&=t(1)-t(0)\\ 0-t(1)&=t(1)-t(0)\\ t(0)&=2t(1)\\ \frac{\pi}{c(2p+\frac12)}\left(2^{2p+\frac12}-0^{2p+\frac12}\right)&= 2\frac{\pi}{c(2p+\frac12)}\left(2^{2p+\frac12}-1^{2p+\frac12}\right)\\ 2^{2p+\frac12}&= 2\left(2^{2p+\frac12}-1\right)\\ 2^{2p+\frac12}&= 2\cdot 2^{2p+\frac12}-2\\ 2&=2^{2p+\frac12}\\ 1&=2p+\frac12\\ p&=\frac14 \end{aligned}$$