Write the rational function given as a partial sum of partial fractions.

$$\frac{4 x^{3}+4 x^{2}+x-1}{x^{2}(x+1)^{2}}$$

The rectangular channel has a width of $8 \mathrm{ft}$ and transports water at $18\left(10^3\right)\ \mathrm{ft}^3 / \mathrm{min}$. If the elevation of the bed drops by $0.75 \mathrm{\ ft}$, find the depth $y_2$ after the depression.

Write the rational function given as a partial sum of partial fractions.

$$\frac{x^{2}-5 x-4}{\left(x^{2}+1\right)(x-3)}$$

The $V_{G S(o n)}$ of an $n$-channel E-MOSFET is
a. Less than the threshold voltage
b. Equal to the gate-source cutoff voltage
c. Greater than $V_{D S(0 \pi)}$
d. Greater than $V_{G S(t h)}$

Hot air at $80^{\circ} \mathrm{C}$ is blown over a $2-m \times 4-m$ flat surface at $30^{\circ} \mathrm{C}.$ If the average convection heat transfer coefficient is $55 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K},$ determine the rate of heat transfer from the air to the plate, in kW.

Solve Laplace’s equation inside a $60^{\circ}$ wedge of radius a subject to the boundary conditions: $(a) u(r, 0) = 0,u\left(r, \frac{\pi}{3}\right)=0,u(a, \theta)=f(\theta)$ $(b) \frac{\partial u}{\partial \theta}(r, 0)=0,\frac{\partial u}{\partial \theta}\left(r, \frac{\pi}{3}\right)=0,u(a, \theta)=f(\theta)$

Two equal and opposite couples of magnitude $M=15 \mathrm{~kN} \cdot \mathrm{m}$ are applied to the channel-shaped beam $AB$. Observing that the couples cause the beam to bend in a horizontal plane, determine the stress (a) at point $C,(b)$ at point $D,(c)$ at point $E$.

Baon Chemicals Unlimited purchases a computer-controlled filter for $100,000. Half of the purchase price is borrowed from a bank at 15 percent compounded annually. The loan is to be paid back with equal annual payments over a 5-year period. The filter is expected to last 10 years, at which time it will have a salvage value of$10,000. Over the 10-year period, the operating and maintenance costs are expected to equal $20,000 in year 1 and increase by$1,500/year each year thereafter. By making the investment, annual fines of $50,000 for pollution will be avoided. Baon expects to earn 12 percent compounded annually on its investments. Based on a present worth analysis, determine whether purchasing the filter is economically justified.

Determine the partial derivative with respect to x and the partial derivative with respect to y at x=4 and y=1 for $f(x, y)=3x^2y^3+2x+3y^6+9$.

Consider a 2-m-high electric hot-water heater that has a diameter of 40 cm and maintains the hot water at $60^\circ C.$ The tank is located in a small room at $20^\circ C$ whose walls and ceiling are at about the same temperature. The tank is placed in a 44-cm-diameter sheet metal shell of negligible thickness, and the space between the tank and the shell is filled with foam insulation. The average temperature and emissivity of the outer surface of the shell are $40^\circ C$ and 0.7, respectively. The price of electricity is $0.08/kWh. Hot-water tank insulation kits large enough to wrap the entire tank are available on the market for about$60. If such an insulation is installed on this water tank by the home owner himself, how long will it take for this additional insulation to pay for itself? Disregard any heat loss from the top and bottom surfaces, and assume the insulation to reduce the heat losses by 80 percent.

Why can the greatest common factor also be called the greatest common divisor?

A long homogeneous resistance wire of radius $r_o = 0.6 cm$ and thermal conductivity $k = 15.2 W/m \cdot K$ is being used to boil water at atmospheric pressure by the passage of electric current, Heat is generated in the wire uniformly as a result of resistance heating at a rate of $16.4 W/cm^3.$ The heat generated is transferred to water at $100^\circ C$ by convection with an average heat transfer coefficient of $h = 3200 W/m^2 K.$ Assuming steady one-dimensional heat transfer, (a) express the differential equation and the boundary conditions for heat conduction through the wire, (b) obtain a relation for the Variation of temperature in the wire by solving the differential equation, and (c) determine the temperature at the centerline of the wire.

In a pharmaceutical plant, a copper pipe $(k_c = 400 W/m \cdot K)$ with inner diameter of 20 mm and wall thickness 15 mm is used for carrying liquid oxygen to a storage tank. The liquid oxygen flowing in the pipe has an average temperature of $−200^\circ C$ and a convection heat transfer coefficient of $120 W/m \cdot K.$ The condition surrounding the pipe has an ambi- ent air temperature of $20^\cdot C$ and a combined heat transter coefficient of $20 W/m^2 \cdot K.$ If the dew point is $10^\circ C,$ determine the thickness of the insulation $(k_i = 0.05 W/m \cdot K)$ around the copper pipe to avoid condensation on the outer surface. Assume thermal contact resistance is negligible.

A $2.2$-mm-diameter and $10$-m-long electric wire is tightly wrapped with a 1-mm-thick plastic cover whose thermal conductivity is $k = 0.15 W/m \cdot K.$ Electrical measurements indicate that a current of 13 A passes through the wire and there is a voltage drop of 8 V along the wire. If the insulated wire is exposed to a medium at $T_{a} = 30^\circ C$ with a heat transfer coefficient of $h = 24 W/m^2 \cdot K,$ determine the temperature at the interface of the wire and the plastic cover in steady operation. Also determine if doubling the thickness of the plastic cover will increase or decrease this interface temperature.