The ideal gas equation relates the volume, pressure, temperature, and thequantity of a

Plot the polynomial y = O.lx5 -0.2x4-x3 + 5x2 -41.5x + 235 in the domain -6 :5:x :5:6 . First create a vector for x, next use the polyval function to calculate y, and then use the plot function.

Plot the polynomial y = 0.008x4-1.8x2-5.4x + 54 in the domain -14 :5: x :5: 16 . First create a vector for x, next use the polyval function to calculate y, and then use the plot function.

Use MATLAB to carry out the following multiplication of two polynomials: (-x3 + 5x -1)(x4 + 2x3 -16x + 5)

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 :S; x :S; 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 ofthree 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 container is 12,212 lb. The specific weight of steel is 0.284 lbfin3.

An aluminum fuel tank has a cylindrical middle section and a semi-spherical ends. The outside diameter is 10 in., and the length of the cylindrical section is 24 in. The wall-thickness of the cylindrical section is t, and the wall-thickness of the semispherical ends is 1.5t. Determine t if the tank weight is 42.27 lb. The specific weight of aluminum is 0.101 lb/in3.

A 20 ft-long rod is cut into 12 pieces, which are welded together to form the frame of a rectangular box. The length of the box's base is 15 in. longer than its width. (a) Create a polynomial expression for the volume V in terms ofx. (b) Make a plot of V versus x. (c) Determine the x that maximizes the volume and determine that volume.

A rectangular piece of cardboard, 40 in. long by 1-40 in. --------1 22 in. wide, is used for making a rectangular box (open top) by cutting out squares of x by x from the corners and folding up the sides. (a) Create a polynomial expression for the volume Vin terms ofx. (b) Make a plot of Vversus x. (c) Determine x if the volume of the box is 1,000 in.3 (d) Determine the value of x that corresponds to the box with the largest possible volume, and determine that volume.

Write a user-defmed function that adds or subtracts two polynomials of any order. Name the function p=polyadd (pll p2 I operation). The ftrst two input arguments pl and p2 are the vectors of the coefficients of the two polynomials. (If the two polynomials are not of the same order, the function adds the necessary zero elements to the shorter vector.) The third input argument operation is a string that can be either 'add 1 or 'sub 1, for adding or subtracting the polynomials, respectively, and the output argument is the resulting polynomial. Use the function to add and subtract the following polynomials: 11 (x) = 2x6- 3x4- 9x3 + llx2- 8x + 4 and 12(x) = 5x3 + 7x -10

Write a user-defmed function that multiplies two polynomials. Name the function p=polymul t (pl1 p2 ). The two input arguments pl and p2 are vectors of the coefficients of the two polynomials. The output argument p is the resulting polynomial. Use the function to multiply the following polynomials: 11 (x) = 2x6- 3x4- 9x3 + llx2- 8x + 4 and 12(x) = 5x3 + 7x -10 Check the answer with MATLAB's built-in function conv.

Write a user-defmed function that calculates the maximum (or minimum) of a quadratic equation of the form: l(x) = ax2+bx+c Name the function [xI y I w] = maxormin (a I b I c) . The input arguments are the coefficients a, b, and c. The output arguments are x, the coordinate of the maximum (or minimum); y, the maximum (or minimum) value; and w, which is equal to 1 if y is a maximum and equal to 2 if y is a minimum. Use the function to determine the maximum or minimum of the following functions: (a) l(x) = 3xL 7x + 14 (b) l(x) = - 5x2 -llx + 15

A cone with base radius r and vertex in contact with the surface of a sphere is constructed inside a sphere, as shown in the figure. The radius of the sphere is R = 9 in. (a) Create a polynomial expression for the volume V of the cone in terms of h. (b) Make a plot of Vversus h for 9 h -9. (c) Using the roots command determine h if the volume of the cone is 500 in. 3. (d) Determine the value of h that corresponds to the cone with the largest possible volume, and determine that volume.

Consider the parabola y = 1.5 (x- 3 )2 + 1 and the Y point P(3, 5.5) . (a) Create a polynomial expression for the distance 6 d from point P to an arbitrary point Q on the parabola. (b) Make a plot of d versus x for 3 x 6 . 4 (c) Determine the coordinates of Q if d = 28 . (d) Determine the coordinates of Q that correspond 2 to the smallest d, and calculate the corresponding value of d.

The following data is given: X 2 5 6 8 9 y 7 8 10 1 1 12 X 2 4 13 15 14 15 (a) Use linear least-squares regression to determine the coefficients m and bin the function y = mx + b that best fits the data. (b) Make a plot that shows the function and the data points.

The boiling temperature of water TB at various altitudes his given in the following table. Determine a linear equation in the form TB = mh + b that best fits the data. Use the equation for calculating the boiling temperature at 5,000 m. Make a plot of the points and the equation.

The U.S. population in selected years between 1815 and 1965 is listed in the table below. Determine a quadratic equation in the form P = a2t2 + at1 + a0 , where t is the number of years after 1800 and P is the population in millions, that best fits the data. Use the equation to estimate the population in 1915 (the population was 98.8 millions). Make a plot of the population versus the year that shows the data points and the equation.

The number of bacteria Ns measured at different times tis given in the following table. Determine an exponential function in the form Ns = Neat that best fits the data. Use the equation to estimate the number of bacteria after 4.5 hr. Make a plot of the points and the equation.

Growth data of a sunflower plant is given in the following table: Week 1 3 5 7 9 11 13 Height (em) 22 51 127 202 227 248 252 The data can be modeled with a function in the form H = C/(1 + Ae-Bt) (logistic equation), where H is the height, Cis a maximum value for H, A and B are constants, and t is the number of weeks. By using the method described in Section 8.2.2, and assuming that C = 254 em, determine the constants A and B such that the function best fit the data. Use the function to estimate the height in week 6. In one figure, plot the function and the data points.

Use the growth data from Problem 22 for the following: (a) Curve-fit the data with a third-order polynomial. Use the polynomial to estimate the height in week 6. (b) Fit the data with linear and spline interpolations and use each interpolation to estimate the height in week 6. In each part make a plot of the data points (circle markers) and the fitted curve or the interpolated curves. Note that part (b) has two interpolation curves.

The following points are given: X 1 2.2 3.7 6.4 9 11.5 14.2 17.8 20.5 23.2 y 12 9 6.6 5.5 7.2 9.2 9.6 8.5 6.5 2.2 (a) Fit the data with a first-order polynomial. Make a plot of the points and the polynomial. (b) Fit the data with a second-order polynomial. Make a plot of the points and the polynomial. (c) Fit the data with a third-order polynomial. Make a plot of the points and the polynomial. (d) Fit the data with an fifth-order polynomial. Make a plot of the points and the polynomial.

The standard air density, D (average of measurements made), at different heights, h, from sea level up to a height of 33 km is given below. h(km) 0 3 6 9 12 15 D (kg/m3) 1.2 0.91 0.66 0.47 0.31 0.19 h(km) 18 21 24 27 30 33 D (kg/m3) 0.12 0.075 0.046 0.029 0.018 0.011 (a) Make the following four plots of the data points (density as a function of height): (1) both axes with linear scale; (2) h with log axis, D with linear axis; (3) h with linear axis, D with log axis; ( 4) both log axes. According to the plots choose a function (linear, power, exponential, or logarithmic) that best fits the data points and determine the coefficients of the function. (b) Plot the function and the points using linear axes.

Write a user-defmed function that fits data points to a power function of the form y = bxm. Name the function [b, m] = powerfit (x,y), where the input arguments x and y are vectors with the coordinates of the data points, and the output arguments b and m are the constants of the fitted exponential equation. Use powerf it to fit the data below. Make a plot that shows the data points and the function. X 0.5 2.4

Viscosity is a property of gases and fluids that characterizes their resistance to flow. For most materials viscosity is highly sensitive to temperature. For gases, the variation of viscosity with temperature is frequently modeled by an equation of the form CT312 Jl = T+S where Jl is the viscosity, Tis the absolute temperature, and C and S are empirical constants. Below is a table that gives the viscosity of air at different temperatures (data from B.R. Munson, D.F. Young, and T.H. Okiishi, Fundamentals of Fluid Mechanics, 4th ed., John Wiley and Sons, 2002).Determine the constants C and S by curve-fitting the equation to the data points. Make a plot of viscosity versus temperature (in o C). In the plot show the data points with markers and the curve-fitted equation with a solid line. The curve fitting can be done by rewriting the equation in the form T312 1 S - = -T+- Jl c c and using a first-order polynomial.

Measurements of the fuel efficiency of a car FE at various speeds v are shown in the table. v (milh) 5 15 25 35 45 55 65 75 FE(mpg) 11 22 28 29.5 30 30 27 23 (a) Curve-fit the data with a second-order polynomial. Use the polynomial to estimate the fuel efficiency at 60 milh. Make a plot of the points and the polynomial. (b) Curve-fit the data with a third-order polynomial. Use the polynomial to estimate the fuel efficiency at 60 milh. Make a plot of the points and the polynomial. (c) Fit the data with linear and spline interpolations. Estimate the fuel efficiency at 60 milh with linear and spline interpolations. Make a plot that shows the data points and curves made of interpolated points.

The relationship between two variables P and t is known to be: p = mt b+t The following data points are given t 8 10 p 6.4 6.6 Determine the constants m and b by curve-fitting the equation to the data points. Make a plot of P versus t. In the plot show the data points with markers and the curve-fitted equation with a solid line. (The curve fitting can be done by writing the reciprocal of the equation and using a first-order polynomial.)

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 force, as a function of elongation, that was required to stretch a rubber specimen that was initially 3 in. long is displayed in the following table. (a) Curve-fit the data with a forth-order polynomial. Make a plot of the data points and the polynomial. Use the polynomial to estimate the force when the rubber specimen was 11.5 in. long. (b) Fit the data with spline interpolation (use MATLAB's built-in function interpl). Make a plot that shows the data points and a curve made by interpolation. Use interpolation to estimate the force when the rubber specimen was 11.5 in. long.

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 Hall-Petch equation: a = a + kj) y 0 The following are results from measurements of average grain diameter and yield stress. (a) Using curve fitting, determine the constants cr0 and k in the Hall-Petch equation for this material. Using the constants determine with the equation the yield stress of material with a grain size of 0.05 mm. Make a plot that shows the data points with circle markers and the curve derived from the Hall-Petch equation with a solid line. (b) Use linear interpolation to determine the yield stress of material with a grain size of 0.05 mm. Make a plot that shows the data points with circle markers and the linear interpolation with a solid line. (c) Use cubic interpolation to determine the yield stress of material with a grain size of 0.05 mm. Make a plot that shows the data points with circle markers and cubic interpolation with a solid line.

The transmission of light through a transparent solid can be described by the equation: IT = Io(l - R)2e-l3L where Ir is the transmitted intensity, I0 is the intensity of the incident beam, B is the absorption coefficient, L is the length of the transparent solid, and R is the fraction of light which is reflected at the interface. If the light is normal to the interface and the beams are transmitted through air, R = (: D 2 where n is the index of refraction for the transparent solid. Experiments measuring the intensity of light transmitted through specimens of a transparent solid of various lengths are given in the following table. The intensity of the incident beam is 5 watts/m2. L(cm) 0.5 1.2 1.7 2.2 4.5 6.0 I r ( watts/m2) 4.2 4.0 3.8 3.6 2.9 2.5 Use this data and curve fitting to determine the absorption coefficient and index of refraction of the solid.

The ideal gas equation relates the volume, pressure, temperature, and the quantity of a gas by: V = nRT p where Vis the volume in liters, P is the pressure in atm, Tis the temperature in kelvins, n is the number of moles, and R is the gas constant. An experiment is conducted for determining the value of the gas constant R. In the experiment, 0.05 mol of gas is compressed to different volumes by applying pressure to the gas. At each volume, the pressure and temperature of the gas are recorded. Using the data given below, determineR by plotting V versus TIP and fitting the data points with a linear equation. V (L) 0.75 0.65 0.55 0.45 0.35 T (C) 25 37 45 56 65 P (atm) 1.63 1.96 2.37 3.00 3.96