When the growth of a spherical cell depends on the flow of nutrients through the surface, it is reasonable to assume that the growth rate, $d V / d t$, is proportional to the surface area, S. Assume that for a particular cell $d V / d t=\dfrac{1}{3} \cdot S$. At what rate is its radius $r$ increasing?

A 5-cm-diameter horizontal jet of water with a velocity of 40 m/s relative to the ground strikes a flat plate that is moving in the same direction as the jet at a velocity of 10 m/s. The water splatters in all directions in the plane of the plate. How much force does the water stream exert on the plate?

A cell receives nutrients through its surface, and its surface area is proportional to the two-thirds power of its weight. Therefore, if w(t) is the cell's weight at time t, then w(t) satisfies w'=a w$^{2 / 3}$, where a is a positive constant. Solve this differential equation with the initial condition w(0)=1 (initial weight 1 unit).

Is a student identification number an example of a permutation or a combination?

Two tomato strains, A and B, both produce fruit that weighs, on average, $1 \mathrm{lb}$ each. All of the variance is due to $V_{\mathrm{G}}$. When these two strains are crossed to each other, the $F_1$ offspring display heterosis with regard to fruit weight, with an average weight of 2 $\mathrm{lb}$. You take these $\mathrm{F}_1$ offspring and backcross them to strain A. You then grow several plants from this cross and measure the weights of their fruit. What would be the expected results for each of the following scenarios?
A. Heterosis is due to a single overdominant gene.
B. Heterosis is due to two dominant genes, one in each strain.
C. Heterosis is due to two overdominant genes.
D. Heterosis is due to dominance of several genes each from strains $\mathrm{A}$ and $\mathrm{B}$.

Compute the deoxygenation rate constant and reaeration rate constant (base e) for the following wastewater and stream conditions.

$$\begin{matrix} \text{ } & \text{k} & \text{Temperature} & \text{H} & \text{Velocity} & \text{ }\\ \text{Source} & \ {\left(\mathrm{day}^{-1}\right)} & \ {\left(^{\circ} \mathbf{C}\right)} & \text{(m)} & \ {\left(\mathbf{m} \cdot \mathbf{s}^{-1}\right)} & \ {\eta}\\ \text{Wastewater} & \text{0.20} & \text{20} & \text{ } & \text{ } & \text{ }\\ \text{Amaunet Stream} & \text{ } & \text{20} & \text{3.0} & \text{0.5} & \text{0.4}\\ \end{matrix}$$

Answer: $k_{d}=0.27 \mathrm{day}^{-1} ; k_{r}=0.53 \mathrm{day}^{-1}$

In the earlier example given, was treated the merry-go-round plus child as a system (shown in the given figure) whose angular momentum is conserved. The merry-go-round is initially at rest, so its initial angular momentum is zero, but at the end it is rotating, so its final angular momentum is nonzero. The angular momentum of the merry-go-round alone thus changes, which can only happen if there is a torque on the merry-go-round. What force provides this torque?

(a) The force of gravity on the merry-go-round

(b) The force of air drag on the merry-go-round

(c) The force produced by the child when she jumps off

(d) The force of friction at the axle of the merry-go-round

The Stanford linear accelerator can accelerate electrons to kinetic energies of $50 \mathrm{GeV}$. What's the de Broglie wavelength of these electrons? Compare with the diameter of a proton, about $2 \mathrm{fm}$.

A point source with volume flow Q = 30 m³/s is immersed in a uniform stream of speed 4 m/s. A Rankine half-body of revolution results. Compute (a) the distance from source to the stagnation point and (b) the two points (r, θ) on the body surface where the local velocity equals 4.5 m/s.

The A-36 steel bolt is tightened within a hole so that the reactive torque on the shank AB can be expressed by the equation

$$t = (kx^{2/3}) N · m/m,$$

where x is in meters. If a torque of T = 50 N · m is applied to the bolt head, determine the constant k and the amount of twist in the 50-mm length of the shank. Assume the shank has a constant radius of 4 mm.

In the half-reaction $\mathrm{NO}_{3}^{-} \rightarrow \mathrm{NH}_{4}^{+}$, on which side of the equation should electrons be added? Add the correct number of electrons to the side on which they are needed, and rewrite the equation.

A cylindrical pressure vessel having a radius $r=14\ \text{in.}$ and wall thickness $t=0.5\ \text{in.}$ is subjected to internal pressure $p=375\ \text{psi}$. In addition, a torque $T=90\ \mathrm{kip}- \mathrm{ft}$ acts at each end of the cylinder. (a) Determine the maximum tensile stress $\sigma_{\max }$ and the maximum in-plane shear stress $\tau_{\operatorname{mix}}$ in the wall of the cylinder. (b) If the allowable in-plane shear stress is $4.5\ \mathrm{ksi}$, what is the maximum allowable torque $T$ ? (c) If $T=150\ \mathrm{kip-ft}$ and allowable in-plane shear and allowable normal stresses are $4.5\ \mathrm{ksi}$ and $11.5\ \mathrm{ksi}$, respectively, what is the minimum required wall thickness?

"Suppose the snark population on Citsigol Island is currently 3,400 and growing at a rate of 18% per year.

a. Write a recursive definition for the sequence $P$ of snark populations $n$ years from now (assuming an unlimited growth model).

b. Write an explicit formula for the population $P$ of snarks $n$ years from now (assuming an unlimited growth model).

c. If the snark population is modeled with the limited growth model difference equation $P_{n+1}=P_n+0.18 P_n\left(1-\frac{P_n}{17000}\right)$, what is the end behavior of the sequence $P$ ? What does the end behavior mean in this situation?"

We all carry 20,000 to 25,000 genes in our genome. So far, patents have been issued for more than 6000 of these genes. Do you think that companies or individuals should be able to patent human genes? Why or why not?