DESCRIPTION: The growth of a fish could be modeled with an ordinary differential equation (1): dtdw=awnbw In equation (1), w is the weight of the fish [lb],a=5.05[bb(1) week 1],b=2[ week 1], and n is a real number that can be interpreted as 'feeding efficiency' and may depend on many parameters (including the amount of food available to the fish, fish's health, and temperature). A local fish farmer purchased some young fish and released them into her pond to grow. The initial weight of the fish at t0=0 was w0=0.5[lb]. The farmer has the ability to control the feeding efficiency n within a narrow range 0.6n0.7 and knows from previous experience that the fish grow to their maximum weight in about 20 weeks no matter what. The demand on the fish market is such that the farmer profits most from selling fish that weigh about 10lbs each. The farmer is interested in the time, t1 [weeks], it would take the fish to grow to the weight of w1 =10 [lb], for every given value of feeding efficiency n (maintaining high feeding efficiency is expensive). The analytical solution to this problem is given by equation (2): t1=b(n1)1ln(bw0(1n)abw1(1n)a) The farmer could not solve equation (1) to find t1 analytically, and measured the dependence of t1. on the feeding efficiency n experimentally instead: SIGNMENT: (1) [5 points] Plot the dependence of t1 [weeks] on n [unitless] obtained experimentally (data from the table). (2) Obtain the dependence of t1 [weeks] on n [unitless] numerically, and plot the numerical solution on the same plot. (You must complete both option (a) and option (b) to receive full credit. You cannot use array fun or find functions to solve this problem.) a. [10 points] Solve equation (1) using an ODE solver (note that 0t20 and w0 =0.5 ) and use the solution to find the time t1 [weeks] it would take the fish to reach the target weight w1=10 (lbs) for every value of n b. [10 points] Re-write equation (1) as dt=awnbwdw and solve it by integrating the left-hand side from t0=0 to t1 (weeks) and the right-hand side from w0 =0.5[lbs] to w1=10[lbs] numerically (that is, t1=w0w1awnbw1dw ) for every value of n. HINT: For each option, first create an m-file function that finds t1 for a single value of n, then execute this function for every value of n using the programming skills you acquired in ENG1331. (3) [5 points] On the same graph, plot the analytical solution given by equation (2). Make the numerical and the analytical solutions look distinct on the graph (yes, all three of them). (4) [10 points] Fit the experimental data with a reciprocal function, y=mx+b1. In a new figure, plot the data and the fitted curve. Find t1 (weeks) for n=0.653 (usitig the reciprocal fit). (5) [10 points] Interpolate the experimental data with 'cubic' method, and plot the interpolant in the same figure. Find t1 (weeks) for n=0.653 (using cubic interpolation). Format of report submission: - Create 'Lastname_Firstname_Exam1.m' m-file: \% Lastname, Firstname % Exam \#1 about fish % Date clear all close all clc disp('Problem 1 start:') disp('Problem 1 end') disp(" disp('Problem 2 start:') disp('Problem 2 end')