Question 1. For several values of x, use MATLAB to con rm that sinh x = (e^x - e^-x)/2.

Question 2. For several values of x, use MATLAB to con rm that cosh^-1x = In (x + √(x² - 1)

Question 3. The capacitance of two parallel conductors of length L and radius r, separated by a distance d in air, is given by

C= Π∈L/ln[(d - r)/r]

where ∈ is the permittivity of air (∈ = 8.854 X 10^-12 F/m).

Write a script le that accepts user input for d, L, and r and computes and displays C. Test the le with the values L = 1 m, r = 0.001 m, and d = 0.004 m.

Question 4. The output of the MATLAB atan2 function is in radians. Write a func-tion called atan2d that produces an output in degrees.
9. Write a function that accepts temperature in degrees Fahrenheit (°F) and computes the corresponding value in degrees Celsius (°C). The relation between the two is

T°C = 5/9 (T°F - 32)

Be sure to test your function.

Question 5. An object thrown vertically with a speed v_o reaches a height h at time t, where

h = v_ot - 1/2gt²

Write and test a function that computes the time t required to reach a speci ed height h, for a given value of vo. The function's inputs should be h, vo, and g. Test your function for the case where h = 100 m, v_o = 50 m/s, and g = 9.81 m/s². Interpret both answers.

Question 6. A water tank consists of a cylindrical part of radius r and height h and a hemispherical top. The tank is to be constructed to hold 600 m³ when lied. The surface area of the cylindrical part is 2Πrh, and its volume is Πr²h. The surface area of the hemispherical top is given by 2Πrh, and its volume is given by Πr²h. The cost to construct the cylindrical part of the tank is $400 per square meter of surface area; the hemispherical part costs $600 per square meter. Use the fminbnd function to compute the radius that results in the least cost. Compute the corresponding height h.

Question 7. A fence around a eld is shaped as shown in Figure P12. It consists of a rectangle of length L and width W, and a right triangle that is symmetrical about the central horizontal axis of the rectangle. Suppose the width W is known (in meters) and the enclosed area A is known (in square meters). Write a user-de ned function le with W and A as inputs. The outputs are the length L required so that the enclosed area is A and the total length of fence required. Test your function for the values W = 6 m and A = 80 m².

Question 8. A fenced enclosure consists of a rectangle of length L and width 2R and a semicircle of radius R, as shown in Figure P13. The enclosure is to be built to have an area A of 2000 ft². The cost of the fence is $50 per foot for the curved portion and $40 per foot for the straight sides. Use the fminbnd function to determine with a resolution of 0.01 ft the values of R and L required to minimize the total cost of the fence. Also compute the minimum cost.

Question 9. Using estimates of rainfall, evaporation, and water consumption, the town engineer developed the following model of the water volume in the reser-voir as a function of time

V(t) = 10⁹ + 10⁸(1 - e^-t/100) - rt

where V is the water volume in liters, t is time in days, and r is the town's consumption rate in liters per day. Write two user-de ned functions. The first function should de ne the function V(t) for use with the f zero func-tion. The second function should use f zero to compute how long it will take for the water volume to decrease to x percent of its initial value of 10⁹L.

The inputs to the second function should be x and r. Test your functions for the case where x = 50 percent and r = 10⁷ L/day.