Using python Solve the following equation numerically using the Heun method using  and    with the initial condition at   As part of the solution process, converge the solution by decreasing the step size .  To determine if the solution is converged, you can find the maximum  for each step size  and plot maximum  .vs decreasing stepsize .

(Mechanics) The deflection at any point along the centerline of a cantilevered beam, such as the one used for a balcony (see Figure 5.15), when a load is distributed evenly along the beam is given by this formula: d=wx224EI(x2+6l24lx) d is the deflection at location x (ft). xisthedistancefromthesecuredend( ft).wistheweightplacedattheendofthebeam( lbs/ft).listhebeamlength( ft). Eisthemodulesofelasticity( lbs/f t 2 ).Iisthesecondmomentofinertia( f t 4 ). For the beam shown in Figure 5.15, the second moment of inertia is determined as follows: l=bh312 b is the beam’s base. h is the beam’s height. Using these formulas, write, compile, and run a C++ program that determines and displays a table of the deflection for a cantilevered pine beam at half-foot increments along its length, using the following data: w=200lbs/ftl=3ftE=187.2106lb/ft2b=.2fth=.3ft

(Numerical) Determine and write an algorithm to sort four numbers into ascending (from lowest to highest) order.

(Numerical analysis) Here’s a challenging problem for those who know a little calculus. The Newton-Raphson method can be used to find the roots of any equation y(x)=0. In this method, the (i+1)stapproximation,xi+1,toarootofy(x)=0 is given in terms of the ith approximation, xi, by the following formula, where y’ denotes the derivative of y(x) with respect to x: xi+1=xiy(xi)/y(xi) For example, if y(x)=3x2+2x2,theny(x)=6x+2 , and the roots are found by making a reasonable guess for a first approximation x1 and iterating by using this equation: xi+1=xi(3xi2+2xi2)/(6xi+2) a. Using the Newton-Raphson method, find the two roots of the equation 3x2+2x2=0. (Hint: There’s one positive root and one negative root.) b. Extend the program written for Exercise 6a so that it finds the roots of any function y(x)=0, when the function for y(x) and the derivative of y(x) are placed in the code.

(Statics) A beam’s second moment of inertia, also known as its area moment of inertia, is used to determine its resistance to bending and deflection. For a rectangular beam (see Figure 6.6), the second moment of inertia is given by this formula: Ibh3/12 I is the second moment of inertia (m4). b is the base (m). h is the height (m). a. Using this formula, write a function called beamMoment() that accepts two double- precision numbers as parameters (one for the base and one for the height), calculates the corresponding second moment of inertia, and displays the result. b. Include the function written in Exercise 4a in a working program. Make sure your function is called from main(). Test the function by passing various data to it.

One way to determine how healthy a person is by measuring the body fat of the person. The formulas to determine the body fat for female and male are as follows: Body fat formula for women: A1 = (body weight x 0.732) + 8.987 A2 = wrist measurement (at fullest point)/3.140 A3 = waist measurement (at navel) x 0.157 A4 = hip measurement (at fullest point) x 0.249 A6 = forearm measurement (at fullest point) x 0.434 B = A1 + A2 A3 A4 + A5 Body fat = body weight - B Body fat percentage = body fat x 100/body weight Body fat formula for men: A1 = (Body weight x 1.082) + 94.42 A2 = wrist measurement x 4.15 B = A1 A2 Body fat = body weight B Body fat percentage = body fat x 100/body weight Write a program to calculate the body fat of a person.

(Transportation) Road construction requires estimating the expected loads on a road’s pavement over its design life. A common approach for determining this information is to use ESAL values; one ESAL is the load a single 18,000-lb (80,000 N) single-axle truck applies to the road’s surface. The ESAL value for any single-axle vehicle can be approximated by this formula: ESAL=[W18,000]4 ESAL is the equivalent single-axle load. W is the vehicle’s weight (lbs). Using this formula, write, compile, and run a C++ program that determines ESAL values and use it to complete the following chart. The ESAL values should be output in a field width of 10, with six digits after the decimal point.

(Desk check) Determine the value in total after each of the following loops is executed: a.total=0;for( i=1;i=10;i=i+1)total=total+1;b.total=1;for( count=1;count=10;count=count+1)total=total2;c.total=0;for( i=10;i=15;i=i+1)total=total+i;d.total=50;for( i=1;i=10;i=i+1)total=totali;e.total=1;for( icnt=1;icnt=8;++icnt)total=totalicnt;f.total=1.0;for( j=1;j=5;++j)total=total/2.0;

(Debug) Determine and correct the errors in the following programs. a.includeiostreamusingnamespacestd;intmain()width=15area=lengthwidth;coutTheareaisareab.includeiostreamusingnamespacestd;intmain()intlength,width,area;area=lengthwidth;length=20;width=15;coutTheareaisarea;return0;c.includeiostreamintmain()intlength=20;width=15,area;lengthwidth=area;coutTheareais,area;return0;

(Computation) Among other applications, Pascal’s triangle (see Figure 7.22) provides a means of determining the number of possible combinations of n things taken r at a time. For example, the number of possible combinations of five people (n = 5) taken two at a time (r=2)is10. Each row of the triangle begins and ends with 1. Every other element in a row is the sum of the element directly above it with the element to the left of the one above it. That is, element[n][r]=element[n1][r]+element[n1][r1] Using this information, write and test a C++ program to create the first 11 rows of a twodimensional array representing Pascal’s triangle. For any given value of n less than 11 and r less than or equal to n, the program should display the correct element. Use your program to determine in how many ways a committee of 8 can be selected from a group of 10 people