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Questions and Answers of Finance

Use MATLAB to carry out the following multiplication of two polynomials: (-x3 + 5x - 1)(x4 + 2x3 -16x + 5)
When rubber is stretched, its elongation is initially proportional to the applied force, but as it reaches about twice its original length, the force required to stretch the rubber increases rapidly.
The yield strength, cry, of many metals depends on the size of the grains. For these metals, the relationship between the yield stress and the average grain diameter d can be modeled by the
The transmission of light through a transparent solid can be described by the equation:IT = I0(l - R)2e-BLwhere IT is the transmitted intensity, I0 is the intensity of the incident beam, B is the
The ideal gas equation relates the volume, pressure, temperature, and the quantity of a gas by:V = nRT/Pwhere Vis the volume in liters, P is the pressure in atm, T is the temperature in kelvins, n is
Use MATLAB to carry out the following multiplication of polynomials: x(x - 1.7)(x + 0.5)(x - 0.7)(x + 1.5) Plot the polynomial for -1.6 < x < 1.8.
Divide the polynomial -10x6 - 20x5 + 9x4 + 10x3 + 8x2 + 11x - 3 by the polynomial 2x2 + 4x - 1.
Divide the polynomial -0.24x7 + 1.6x6 + 1.5x5 - 7.41x4 - 1.8x3 - 4x2 - 75.2x - 91 by the polynomial -0.8x3 + 5x + 6.5.
The product of two consecutive integers is 6,972. Using MATLAB's built-in function for operations with polynomials, determine the two integers.
The product of three integers with spacing of 5 between them (e.g., 9, 14, 19) is 10,098. Using MATLAB's built-in function for operations with polynomials, determine the three integers.
A rectangular steel container has the outside dimensions shown in the figure The thickness of the bottom and top walls is t, and the thickness of side walls is t/2. Determine t if the weight of the
Determine the solution of the equation e0.3x - x2 = -4.
A paper cup shaped as a frustum of a cone with R2 = 2R1 is designed to have a volume of 250 cm3. Determine R1, R2, and h such that the least amount of paper will be used for making the cup. The
Consider again the block that is being pulled in Problem 5. Determine the angle e at which the force that is requires to pull the box is the smallest. What is the magnitude of this force?I problem 5A
Determine the dimensions (radius r and height h) and the volume of the cylinder with the largest volume that can be made inside of a sphere with a radius R of 14 in.
Consider the ellipse x2/192 + y2/52 = 1. Determine the sides a and b of the rectangle with the largest area that can be enclosed by the ellipse.
Planck's radiation law gives the spectral radiancy R as a function of the wave length A and temperature T (in kelvins):where c = 3.0 Ã— 108 m/s is the speed of light, h = 6.63 Ã— 10-34 J s is
A 108 in.-1ong beam AB is attached to the wall with a pin at point A and to a 68 in.€“long cable CD. A load W = 250 lb is attached to the beam at point B. The tension in the cable T is given
Use MATLAB to calculate the following integral:(a)(b)
Use MATLAB to calculate the following integrals:(a)(b)
The speed of a race car during the first seven seconds of a race is given by:Determine the distance the car traveled during the first six seconds.
The shape of the centroid line of the Gateway Arch in St. Louis can be modeled approximately with the equation:f(x) = 693.9 - 68.8cosh(€“x/99.7) for-299.25 By using the equation:determine the
Determine the solution of the equation 2 cosx - 0.5√x = 1.
The flow rate Q (volume of fluid per second) in a round pipe can be calculated by:For turbulent flow the velocity profile can be estimated by: v = vmax(1 €“ r/R)l/n. Determine Q for R = 0.25 in.,
The electric field E due to a charged circular disk at a point at a distance z along the axis of the disk is given bywhere Ïƒ is the charge density, Îµo is the permittivity constant, Îµo = 8.85
The length of a curve given by a parametric equation x(t), y(t) is given by:The cardioid curve shown in the figure is given by:x = 2bcost - bcos2t, and y = 2bsint - bsin2t with 0
The variation of gravitational acceleration g with altitude y is given byg = R2/(R + y)2 g0where R = 6371 km is the radius of the earth, and g0 = 9.81 m/s2 is the gravitational acceleration at sea
A cross section of a river with measurements of its depth at intervals of 40 ft: is shown in the figure. Use numerical integration to estimate the cross-sectional area of the river.
An aprn-n-w-im11te map of 1he state of Texas is shown in the figure. For determining the area of the state, the map is divided into two parts (one above and one below the x axis). Determine the area
A cross-sectional area has the geometry of half an ellipse, as shown in the figure to the right. The coordinate of the centroid of the area can be calculated by:= My/Awhere A is the area given by A =
The orbit of Pluto is elliptical in shape, with a = 5.9065 Ã— 109k:m. and b = 5.7208 Ã— 109km. The perimeter of an ellipse can be calculated bywhere k = ˆša2 €“ b2/a. Determine the
The Fresnel integrals are:Calculate S(x) and C(x) for 0
Use a MATLAB built-in function to numerically solve: dy/dx = 2x/3y2 for 1 < / x < 5 with y(1) = 2
Determine the two roots of the equation x3- 5x2.5 + e0.9x + 4(x + 1) = -2.
Use a MATLAB built-in function to numerically solve: dx/dy = 2x + 1/y + 2 for 0 < x < 8 with y(0) = 2
Use a MATLAB built-in function to numerically solve: dy/dt = 80e-1.6tcos(4t) - 0.4y for 0 < x < 5 with y(1) = 1
Use a MATLAB built-in function to numerically solve: dy/dx = –x2 + x3e–y/4 for 1 < x < 5 with y(1) = 1
The growth of a fish is often modeled by the von Bertalanffy growth model: dw/dt = aw2/3 - bw where w is the weight and a and b are constants. Solve the equation for w for the case a = 5/b1/3, b = 2
A water tank shaped as an ellipsoid (a = 1.5 m, b = 4.0 m, c = 3m) has a circular hole at the bottom, as shown. According to Torricelli's law, the speed v of the water that is discharging from the
The sudden outbreak of an insect population can be modeled by the equationThe first term relates to the well-known logistic population growth model where N is the number of insects, R is an intrinsic
An airplane uses a parachute and other means of braking as it slows down on the runway after landing. Its acceleration is given by a = -0.0035v2 -3 m/s2. Since a = dv/dt, the rate of change of the
The population growth of species with limited capacity can be modeled by the equation: dN/dt = kN(NM - N) where N is the population size, NM is the limiting number for the population, and k is a
An RL circuit includes a voltage source vs, a resistor R = 1.8 n, and an inductor L = 0.4 H, as shown in the figure. The differential equation that describes the response of the circuit is L/R diL/dt
Tumor growth can be modeled with the equationwhere A (t) is the area of the tumor and a , k, and u are constants. Solve the equation for 0
Determine the positive roots of the equation x2- 5x sin(3x) + 3 = 0.
The velocity of an object that falls freely due to the earth gravity can be modeled by the equation: m dv/dt = –mg + kv2 where m is the mass of the object, g = 9.81 m/s2, is and k is a constant.
A box of mass m = 25 kg is being pulled by a rope. The force that is required to move the box is given by:F = Î¼mg/cosÎ¸ + Î¼sinÎ¸where Î¼ = 0.55 is the friction coefficient and g = 9.81
A scale is made of two springs, as shown in the figure. The springs are nonlinear such that the force they apply is given by FS = K1u + K2u3, where the K's are constants and u = L - L0 is the
An estimate of the minimum velocity required for a round flat stone to skip when it hits the water is given by (Lyderic Bocquet, "The Physics of Stone Skipping," Am. J. Phys., vol. 71, no. 2,
The diode in the circuit shown is forward biased. The current I flowing through the diode is given by:I= Is (eqvd/ekt - 1)where vD is the voltage drop across the diode, T is the temperature in
Determine the minimum and the maximum of the function f(x)= 3(x - 0.25)/1 + 3.5(0.8x - 0.3)2
The position of a moving particle as a function of time is given by:Plot the position of the particle for 0
Molecules of a gas in a container are moving around at different speeds. Maxwell's speed distribution law gives the probability distribution P(v) as a function of temperature and speed:where M is the
Plank's distribution law gives the blackbody emissive power (amount of radiation energy emitted) as a function of temperature and wave length:where C1 = 3.742 Ã— 108 WÎ¼m4/m2, C2 = 1.439 Ã—
The flow Q (m3/s) in a rectangular channel is given by the Manning's equation:where d is the depth of water (m), w is the width of the channel (m), S is the slope of the channel (m/m), n is the
An RLC circuit with an alternating voltage source is shown. The source voltage vs is given by vs = vmsin(Ï‰dt), where Ï‰d = 2Ï€fd, in which fd is the driving frequency. The amplitude of the
A defect in a crystal lattice where a row of atoms is missing is called an edge dislocation. The stress field around an edge dislocation is given by:where G is the shear modulus, b is the Burgers
The current I flowing through a semiconductor diode is given byI = Is(eqvd/kt €“ 1)where Is = 10-12 A is the saturation current, q = 1.6 Ã— 10-19 C is the elementary charge
The equation for the streamlines for uniform flow over a cylinder is ψ(x,y) = y – y/x2 + y2 where ψ is the stream function. For example, if ψ = 0, then y = 0. Since the equation is satisfied for
The deflection w of a clamped circular membrane of radius r d subjected to pressure P is given by (small deformation theory)where r is the radial coordinate, and K = Et3/12(1 €“ v2), where E, t,
The Verhulst model, given in the following equation, describes the growth of a population that is limited by various factors such as overcrowding and lack of resources:where N(t) is the number of
The geometry of a ship hull (Wigley hull) can be modeled by the equationwhere x, y, and z are the length, width, and height, respectively. Use MATLAB to make a 3-D figure of the hull as shown. Use B
An elliptical staircase that decreases can be modeled by the parametric equationsx = rcos(t) y = rsin(t) z = ht/Ï€nwhere r = ab/ˆš[bcos(t)2 + [asin(t)2, a and b are the semimajor and semiminor
The stresses fields near a crack tip of a linear elastic isotropic material for mode I loading are given by:For K1 = 300 ksiˆšin plot the stresses (each in a separate figure) in the domain 0
A ball thrown up falls back to the floor and bounces many times. For a ball thrown up in the direction shown in the figure, the position of the ball as a function of time is given by:The velocities
The ladder of a fire truck can be elevated (increase of angle É¸), rotated about the z axis (increase of angle Î¸), and extended (increase of r). Initially the ladder rests on the truck (É¸
Make a 3-D surface plot of the function z = y2/4 - 2sin(l.5x) in the domain -3 < x < 3 and-3 < y < 3.
Make a 3-D surface plot of the function z = 0.5x2 + 0.5y2 in the domain -2 < x < 2 and -2 < y < 2.
Make a 3-D mesh plot of the function z = -cos(2R)/e0.2R, where R = √x2 + y2 in the domain -5 < x < 5 and -5 < Y < 5.
Make a 3-D surface plot of the function z = cos(x)cos(√x2 + y2)e-|0.2x| in the domain -2π < x < 2π and -π < y < π.
Make a plot of the ice cream cone shown in the figure. The cone is 8 in. tall with a 4 in. diameter base. The ice cream at the top is a 4 in. diameter hemisphere.A parametric equation for the cone
The vander Waals equation gives a relationship between the pressure p (atm), volume V, (L), and temperature T (K) for a real gas: p = nRT/V – n2a/V2 where n is the number of moles, R = 0.08206 (L
Define x as a symbolic variable and create the two symbolic expressions S1 = x2(x - 6) + 4(3x - 2) and S2 = (x + 2)2 - 8x Use symbolic operations to determine the simplest form of each of following
Consider the two ellipses in the x y plane given by the equations(a) Use the ezplot command to plot the two ellipses in the same figure.(b) Determine the coordinates of the points where the ellipses
A 120 in.-long beam AB is attached to the wall with a pin at point A and to a 66 in long cable CD. A load W = 200 lb is attached to the beam at point B. The tension in the cable T and the x and y
A box of mass m is being pulled by a rope as shown. The force Fin the rope as a function of x can be calculated from the equations:where N and Î¼ are the normal force and friction coefficient
The mechanical power output Pin a contracting muscle is given by:where T is the muscle tension, v is the shortening velocity (max of vmax), T0 is the isometric tension (i.e., tension at zero
The equation of a circle is xz + yz = Rz, where R is the radius of the circle. Write a program in a script file that first derives the equation (symbolically) of the tangent line to the circle at the
A tracking radar antenna is locked on an airplane flying at a constant altitude of 5 km, and a constant speed of 540 km/h. The airplane travels along a path that passes exactly above the radar
Evaluate the following indefinite integrals:(a)(b)
Define x as a symbolic variable and create the symbolic expressionS = cos2x/1 + sin2xPlot S in the domain 0
The parametric equations of an ellipsoid are: x = a cosusinv, y = b sinusinv, z = c cosv where 0 Show that the differential volume element of the ellipsoid shown is given by:dV = -nabcsin3vdvUse
The one-dimensional diffusion equation is given by:
Define y as a symbolic variable and create the two symbolic expressions S1 = x(x2 + 6x + 12) + 8 and S2 = (x - 3)2 + l0x - 5 Use symbolic operations to determine the simplest form of each of
A ceramic tile has the design shown in the figure. The shaded area is painted red and the rest of the tile is white. The border line between the red and the white areas follows the equationy = -kx2 +
Show that the location of the centroid yc of the half-circle area shown is given by yc = 4R/3Ï€.The coordinate yc can be calculated by:
For the half-circle area shown in the previous problem, show that the moment of inertia about the x axis, Ix, is given by Ix = 1/8 Ï€R4. The moment of inertia Ix can be calculated by:
The rms value of an AC voltage is defined bywhere T is the period of the waveform.(a) A voltage is given by v(t) = Vcos(Ï‰t). Show that vrms = V/ˆš2 and is independent of Ï‰. (The
The spread of an infection from a single individual to a population of N uninfected persons can be described by the equation dx/dt = -Rx(N + l - x) with initial condition x(0) = N where x is the
The Maxwell-Boltzmann probability density function f(v) is given bywhere m (kg) is the mass of each molecule, v (m/s) is the speed, T(K) is the temperature, and k = 1.38 Ã— 10-23 J/K is
The velocity of a skydiver whose parachute is still closed can be modeled by assuming that the air resistance is proportional to the velocity. From Newton's second law of motion the relationship
A resistor R (R = 0.4Ω) and an inductor L (L = 0.08 H) are connected as shown. Initially, the switch is connected to point A and there is no current in the circuit. At t = 0 the switch is moved
Determine the general solution of the differential equation dy/dx = x4 - 2y/2x Show that the solution is correct. (Derive the first derivative of the solution, and then substitute back into the
Determine the solution of the following differential equation that satisfies the given initial conditions. Plot the solution for 0 < t < 7.
Define x and y as symbolic variables and create the two symbolic expressionsUse symbolic operations to determine the simplest form of S · T. Use the subs command to evaluate the numerical value of
The current, i, in a series RLC circuit when the switch is closed at t = 0 can be determined from the solution of the 2nd-order ODEwhere R, L, and C are the resistance of the resistor, the inductance
Damped free vibrations can be modeled by a block of mass m that is attached to a spring and a dashpot as shown. From Newton's second law of motion, the displacement x of the mass as a function of
Define x as a symbolic variable. (a) Derive the equation of the polynomial that has the roots x = -2, x = -0.5, x = 2, and x = 4.5. (b) Determine the roots of the polynomial f(x) = x6 - 6.5x5 - 58x4
Use the commands from Section 11.2 to show that: (a) sin(4x) = 4sinxcosx - 8sin3xcosx (b) cosxcosy = 1/2[cos(x - y) + cos(x + y)]
Use the commands from Section 11.2 to show that: (a) tan(3x) = 3tanx- tan3x/1- 3tan2x (b) sin(x + y + z) = sinxcosycosz + cosxsinycosz + cosxcosysinz - sinxsinysinz

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