Good afternoon! Please help me understand these problem in my reviewer. I have upcoming exam next week. If its okay to give me solutions that would be great! Thank you and keep safe.

1.

A company introduces a new product for which the number of units sold S is given by the equation below, where t is the time in months. S (t ) = 20 3 - 9 6+ (a) Find the average rate of change of S(t) during the first year. (b).During.what month of the first year does S '(t) equal the average rate of change? ---Select--- v Select- Need January aster It February eBook March April May June Submit A July August September October DETAIL November ALC11M 2.6.017.MI. DecemberFind the length and width of a rectangle that has the given perimeter and a maximum area. Perimeter: 92 meters Find two positive numbers whose sum is 190 and whose product is a maximum. (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) First Second Number, X Number Product, P 50 190 - 50 50(190 - 50) = 7,000 50 190 - 60 60(190 - 60) = 7,800 70 190 - 70 70(190 - 70) = 80 190 - 80 80(190 - 80) = 90 190 - 90 90(190 - 90) = 100 190 - 100 100(190 - 100) = (b) Write the product P as a function of x. P ( X ) = (c) Use calculus to find the critical number of the function in part (b). Then find the two numbers. (Enter your answers as a comma-separated list.) (d) Use a graphing utility to graph the function in part (b) and verify the solution from the graph. 10000 8000 9000 150 8000 6000 8950 100 6000 4000 4000 2000 8900 50 2000 20 40 60 80 100 120 140 160 180 200 8850 O 20 40 60 80 100 120 140 160 180 200 20 40 60 80 100 120 140 160 180 200 -2000 O 0 20 40 60 80 100 120 140 160 180 200 The solution appears to be at x =At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 12 cubic meters per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 2 meters high? (Hint: The formula for the Volume of a cone is L/ =1Ir2h.) h' = m/min A rectangular poster is to contain 882 square inches of print. The margins at the top and bottom of the poster are to be 2 inches, and the margins on the left and right are to be 1 inch. What should the dimensions of the poster be so that the least amount of poster is used? in (smaller value) in (larger value} The height of a ball I seconds after it is thrown upward from a height of 3 meters and with an initial velocity of 24.5 meters per second is t) = 4.9t2 + 24.5r + 3. (a) Verify that f(2) : f(3). rm = E m w = E m (b) According to Rolle's Theorem, what must be the velocity at some time in the interval (2, 3)? SW Find that time. A ladder 10 meters long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of; meter per second. (a) What is the velocity of the top of the ladder when the base is given below? (Round your answers to three decimal places.) 2 meters away from the wall Z Man 6 meters away from the wall E W... 8 meters away from the wall E W... (b) Consider the triangle formed by the side of the house, ladder, and the ground. Find the rate at which the area of the triangle is Changing when the base of the ladder is 6 meters from the wall. E m... (c) Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 6 meters from the wall. E radsecr The height of an object 1: seconds after it is dropped from a height of 150 meters is 30:) : 4.9t4 + 150. (a) Find the average velocity of the object during the first 5 seconds. E m (b) Use the Mean Value Theorem to verify that at some time during the first 5 seconds of fali, the instantaneous velocity equals the average velocity. Find that time. :|5 A point is moving along the graph of the given function at the rate dx/dt. Find dy/dt for the given values of x. y = 2x2 + 5; dx = 5 centimeters per second dt (a) cm/sec (b) x =0 cm/sec (c) x =1 cm/secFind two positive numbers that satisfy the given requirements. The sum of the first number squared and the second number is 57 and the product is a maximum. (first number) (second number)An open box of maximum volume is to be made from a square piece of material, s = 24 inches on a side, by cutting equal squares from the corners and turning up the sides (see figure). xI X' S - 2x (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) Height, x Length and Width Volume, V 1 24 - 2(1) 1[24 - 2(1) ]2 = 484 2 24 - 2(2) 2[24 - 2(2) ]2 = 800 3 24 - 2(3) 3[24 - 2(3) 12 = 4 24 - 2(4) 4[24 - 2(4) 12 = 5 24 - 2(5) 5[24 - 2(5) ]2 = 6 24 - 2(6) 6[24 - 2(6) 12 = Use the table to guess the maximum volume. V = (b) Write the volume V as a function of x. V : 0