Archimedes’ principle of buoyancy states that an object submerged in a fluid is buoyed up by a force equal to the weight of the fluid that is displaced by the object. A rectangular box of $1 \times 2 \times 3$ feet and weighing 384 pounds is dropped into a 100-foot deep freshwater lake. The box begins to sink with a drag due to the water having a magnitude equal to 1/2 the velocity. Calculate the terminal velocity of the box. Will the box have achieved a velocity of 10 feet per second by the time it reaches the bottom? Assume that the density of water is 62.5 pounds per cubic foot.

Archimedes’s principle of buoyancy states that an object submerged in a fluid is buoyed up by a force equal to the weight of the fluid displaced. An experimental, spherically shaped sonobuoy of radius $\dfrac{1}{2}$ m with a mass m kg is dropped into the ocean with a velocity of 10 m/s when it hits the water. The sonobuoy experiences a drag force due to the water equal to one-half its velocity.Write down a differential equation describing the motion of the sonobuoy. Find values of m for which the sonobuoy will sink and calculate the corresponding terminal sink velocity of the sonobuoy. The density of seawater is $\rho_{0}$ = 1.025 kg/L.

(Logistic Model of Population Growth) In 1837, the Dutch biologist Verhulst developed a differential equation to model changes in a population (he was studying fish populations in the Adriatic Sea). Verhulst reasoned that the rate of change of a population P(t) with respect to time should be influenced by growth factors (for example, current population) and also factors tending to retard the population (such as limitations on food and space). He formed a model by assuming that growth factors can be incorporated into a term $aP(t)$ and retarding factors into a term $−bP(t)^2$ with a and b as positive constants whose values depend on the particular population. This led to his logistic equation $P(t)=a P(t)-b P(t)^{2}$. Note that, when $b=0$, this is the exponential model. Solve the logistic model, subject to the initial condition $P(0)= p_0$, to obtain $P(t)=\frac{a p_{0}}{a-b p_{0}+b p_{0} e^{a t}} e^{a t}$. This is the logistic model of population growth. Show that, unlike exponential growth, the logistic model produces a population function P(t) that is bounded above and increases asymptotically toward a/b as $t \rightarrow \infty$. Thus, a logistic model produces a population function that never grows beyond a certain value.

A tank shaped like a right circular cone, vertex down, is 9 feet high and has a diameter of 8 feet. It is initially full of water. (a) Determine the time required to drain the tank through a circular hole with a diameter of 2 inches at the vertex. Take $k =0.6$. (b) Determine the time it takes to drain the tank if it is inverted and the drain hole is of the same size and shape as in (a), but now located in the new (flat) base.

Determine the time it takes to drain a spherical tank with a radius of 18 feet if it is initially full of water, which drains through a circular hole with a radius of 3 inches in the bottom of the tank. Use $k =0.8.$