need help in calculus problems. Thank you!

Math 115 / Exam 1 (August 4. 2023) page 4 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 a feet 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: 1 y . the surface area of a cylinder of radius r and length e is 2Are, . the volume of a cylinder of radius r and length ( is y ATA like $1000y The ATA would like to spend exactly $1000 on one shelter. a. [5 points] Find a formula for y in terms of x. 25 . 4 y = legs, r = = 2 , 1 = y, 40. 2. TE. ; 26. y = Zonxy = roof 2 190 % + 20 KCXy = 10 00 5 y + exy = 500 Y (5+ 7( X ) = 500 y = 500 = 100+ 500 5 TAX 100+ 500 Answer: y = 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)? Volume = zy + 7 76 (3 20 ) y = acy ? + - 7 ( 4 x 2 ) y 8 Answer:page 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 f(x) be a continuous function defined for all real numbers whose derivative is given by f' (x ) = (2x + 1)(x -2)2 (x+ 3) 1/3 Find the x-coordinate(s) of all local extrema of the function f(x). Write "NONE" if the function has no local extrema.b. [4 points] Suppose now that we consider f(x) on the domain [3, oo). Given that f(3) = -5, determine the x-coordinates of any GLOBAL extrema of f(x). If none exist write "NONE". Be sure to justify your answers.Math 115 / Exam 1 (August 4. 2023) page 6 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 ins per minute. Recall that the surface area of a sphere of radius r is S = 4mr and the volume is V = =1/3. Be sure to include units 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: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