Assume that the outside temperature varies as

T(t)=15+5 sin ($\pi$ t / 12)

where $t=0$ is 12 noon. A house is heated to $25^{\circ}$C at t=0 and after that, its temperature y(t) varies according to Newton's Law of Cooling :

dy/dt=-0.1(y(t)-T(t))

Use Earlier Exercise to solve for y(t).

In this exercise, find the solutions of the logistic equation dy/dt=y(4-y) satisfying the initial conditions:

(a) y(0)=1

(b) y(0)=4

(c) y(0)=6

Continuing with the previous exercise, let tank 2 be another tank filled with V$_2$ gallons of water. Assume that the dye solution from tank 1 empties into tank 2 as in the previous figure, mixes instantaneously, and leaves tank 2 at the same rate R. Let c$_2$(t) be the dye concentration in tank 2 at time t.

(a) Explain why c$_2$ satisfies the differential equation

$$\frac{d c_2}{d t}=\frac{R}{V_2}\left(c_1-c_2\right)$$

(b) Use the solution to the previous exercise to solve for $c_2(t)$ if $V_1=300$, $V_2=200, R=50$, and $c_0=10$.

(c) Find the maximum concentration in tank 2.

(d) Plot the solution.

In the earlier example, we calculated the thickness of a glacier assuming the friction at the base of the glacier is constant. In this exercise, we consider cases where the friction varies along the length of the glacier.

(a) Solve the glacier thickness differential equation, and use the equation

$$T \frac{d T}{d x}=\frac{\tau}{\rho g}$$

for T(x) with $\tau(x)=0.3 x(1000-x)$ N/m$^2$ and initial condition T(0)=0.

(b) Sketch the graph of T for $0 \leq x \leq 1000$ m.

Tank 1 is filled with $V_1$ liters of water containing blue dye at an initial concentration of $c_0 \mathrm{~g} / \mathrm{L}$. Water flows into the tank at a rate of RL /min, is mixed instantaneously with the dye solution, and flows out through the bottom at the same rate R. Let $c_1(t)$ be the dye concentration in the tank at time t.

(a) Explain why $c_1$ satisfies the differential equation

$$\frac{d c_1}{d t}=-\frac{R}{V_1} c_1$$

(b) Solve for $c_1(t)$ with $V_1=300 \mathrm{~L}$, R=50, and $c_0=10 \mathrm{~g} / \mathrm{L}$.

In the earlier example, we calculated the thickness of a glacier assuming the friction at the base of the glacier is constant. In this exercise, we consider cases where the friction varies along the length of the glacier.

(a) Solve the glacier thickness differential equation, and use the equation below

$$T \frac{d T}{d x}=\frac{\tau}{\rho g}$$

for T(x) with $\tau(x)=75 x$ N/m$^2$ and initial condition T(0)=0.

(b) Sketch the graph of T for $0 \leq x \leq 1000$ m.

The trough (dimensions in centimeters) is filled with water. At time t=0 (in seconds), water begins leaking through a hole at the bottom of area 4$cm^2$ . Let y(t) be the water height at time t . Find a differential equation for y(t) and solve it to determine when the water level decreases to 60 cm .

Solve the Initial Value Problem.

$$y ^ { \prime } = y ^ { 2 } \sin x , \quad y ( \pi ) = 2$$

Solve for l(t), assuming that R = 500 ohms, L = 4 henries, and V = 20 cos(80) volts.

At time t = 0, a tank of height 5 m in the shape of an inverted pyramid whose cross section at the top is a square of side 2 m is filled with water. Water flows through a hole at the bottom of area

$$0.002 \mathrm { m } ^ { 2 }.$$

Use Torricelli's Law to determine the time required for the tank to empty.

Assume that V(t) = V is constant and I(0) = 0. (a) Solve for I(t). (b) Show that

$$\lim _ { t \rightarrow \infty } I ( t ) = V / R$$

and that I(t) reaches approximately 63% of its limiting value after L/R seconds. (c) How long does it take for I(t) to reach 90% of its limiting value if R = 500 ohms, L = 4 henries, and V = 20 volts?

Let A and B be constants. Prove that if A > 0, then all solutions of

$$\frac { d y } { d t } + A y = B$$

approach the same limit as

$$t \rightarrow \infty.$$

Solve the Initial Value Problem. y' = tan y,

$$y ( \ln 2 ) = \frac { \pi } { 2 }$$

Uranium-238 is a radioactive material with a half-life of 4.468 billion years. With t in billions of years, let M(t) be the mass in grams of a sample of uranium-238 that initially consisted of 100 g. Set up and solve an Initial Value Problem for determining M(t).