Urgent! Please help me with calculus. Thank you!

3. [9 points] The Arbor Transit Authorities (ATA) are designing rain shelters for their bus stops. They decide to place a roof in the shape of half a cylinder on four vertical legs of height y feet. The four legs are placed in a rectangle on the ground with width r fect and length y feet. The costs of production are: . $25 for each 1 foot of the total length of the legs, . $40 for each square foot of the area of the roof. The following formulas may be useful in this problem: . the surface area of a cylinder of radius r and length is 2are, . the volume of a cylinder of radius r and length ( is The ATA would like to spend exactly $1000 on one shelter. a. [5 points] Find a formula for y in terms of z. Answer: = b. |4 points] Suppose we used the above to find a formula for the volume of the covered shelter, denoted by V(x). If the ATA wants to make sure that each of the sides of the rectangle has length at least 5 feet, and the height (that is, y) of the shelter is at least 8 feet. What is the domain of the function V(x)? Answer:4. [10 points] In the following questions, use calculus to justify your answers and show enough evidence to demonstrate that you have found them all. Determine your answers algebraically. a. [6 points] Let /(r) be a continuous function defined for all real numbers whose derivative is given by M(I) = (2r + 1)(x - 2)2 (x + 3)1/3 Find the r-coordinate(s) of all local extrema of the function /(r). Write "NONE" if the function has no local extrema. b. |4 points] Suppose now that we consider /(x) on the domain (3, co). Given that /(3) = -5, determine the r-coordinates of any GLOBAL extrema of /(r). II none exist write "NONE". Be sure to justify your answers.5. [13 points| A snowman was built on the diag, and since it is a bright and sunny day his head starts to melt. The surface area of his head (which is a perfect sphere) decreases at a constant rate of 40 in* per minute. Recall that the surface area of a sphere of radius r is S = 4ar and the volume is V = = ap. Be sure to include unils in your answers. a. [3 points] How fast is the radius of the snowball changing when the radius is 5 inches? Answer: b. |4 points] How fast is the volume changing when the radius is 5 inches? Answer: c. [3 points] Write a formula for V in terms of S and r. Your expression should include both S and r. Answer: V = d. [3 points] Use the above formula to verify the rate at which the surface area is changing when the radius is 5 inches. You should use your answer from parts a. and b