2 Find the largest area of a rectangle with one vertex on the parabola y = 64 x , another at the origin, and the remaining two on the positive xaxis and positive yaxis, respectively (see the gure). (Use symbolic notation and fractions where needed.) Note: The length of the base of the rectangle is x and the height of the rectangle is y . Also, note that the point (x, y) is on the graph of y = 64 x2. You can look at the "Student Hint" for more help. Amax = square units > A cylindrical container that has a capacity of 7 m3 is to be produced. The top and bottom of the container are to be made of a material that costs $12 per square meter, while the side of the container is to be made of a material costing $9 per square meter. Find the dimensions that will minimize the cost of the material. (Use decimal notation. Give your answer to three decimal places.) Recall: Volume of a Cylinder is V = m2}: and the Surface Area of a Cylinder is SA = Zrh + err2 Note: For more help with the set up of this question, click on the "Student Hint". 3 A self-catalytic chemical reaction results in the formation of a product that causes its formation rate to increase. The reaction rate V of one of such reactions is given by V = 1.1x(6 x), 0 g x 5 6, where 6 is the initial amount of the chemical and x is the variable amount of the chemical. For What value of x is the reaction rate maximum? (Use symbolic notation and fractions where needed.) A charter ight club charges its members $280 per year. But for each new member in excess of 50, the charge for every member is reduced by $2. Assuming the club will have at least 50 members, What number of members leads to a maximum revenue? (Give your answer as a whole or exact number.) Note: Let x equal the number of members over 50. Check the "Student Hint" for more help. number of members: , Hector manages Food Plaza, a small grocery store. Based on the previous year's sales data, he estimates that Food Plaza can sell a = 2000p + 7520 lemons this year, where p is the price of a single lemon and a is the total amount of lemons sold at price p. Walt supplies Food Plaza with all the fruits and vegetables they sell. Walt can sell Hector at most 5000 lemons in a single year. If Food Plaza pays Walt $1.70 per lemon on average, how many lemons should Hector buy from Walt in order to maximize the store's prot? Recall: P = R C, where P is prot, R is revenue, and C is cost. R, revenue, is equal to the amount of lemons sold, a, times the price of the lemons, p. Also, assume that Hector will sell all of the lemons he purchases from Walt. Note: For more help on the set up of this question, click on "Student Hint". Give your answer as a Whole number. l A car rental agency has 28 cars (each an identical model). The owner of the agency nds that at a price of $22 per day, all the cars can be rented; however, for each $1 increase in rental cost, one of the cars is not rented. What should the agency charge to maximize income? (Use symbolic notation and fractions Where needed.) Note: Let x be the number of $1 increases in rental price above $22 and the Rental Income is calculated by the price of the rental times the number of cars rented. agency should charge: $ l per day