A company has developed a profit model that depends on x, the number of watches sold...

 

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A company has developed a profit model that depends on x, the number of watches sold per month and the number of hours spent advertising y per month, according to z = f(x, y) = 48x+96y - x - 3xy - 9y, where z is measured in thousands of dollars. The minimum number of watches and maximum number of watches that can be sold a month is 0 and 50 respectively, and the minimum and maximum amount of hours spent advertising is 0 and 25, respectively. (a) Define the domain in which f is defined. (b) Find the values x and y that maximize the profit, and find the maximum profit. Exercise 4 - Lagrange Multipliers. (4 points) Using the method of Lagrange multipliers: (a) Find the area of the largest rectangle, with sides parallel to the coordinate axes that can be inscribed in the ellipse x+4y = 1. (b) Give the coordinates of the corner of the rectangle in the first quadrant, i.e. x > 0, y > 0. A company has developed a profit model that depends on x, the number of watches sold per month and the number of hours spent advertising y per month, according to z = f(x, y) = 48x+96y - x - 3xy - 9y, where z is measured in thousands of dollars. The minimum number of watches and maximum number of watches that can be sold a month is 0 and 50 respectively, and the minimum and maximum amount of hours spent advertising is 0 and 25, respectively. (a) Define the domain in which f is defined. (b) Find the values x and y that maximize the profit, and find the maximum profit. Exercise 4 - Lagrange Multipliers. (4 points) Using the method of Lagrange multipliers: (a) Find the area of the largest rectangle, with sides parallel to the coordinate axes that can be inscribed in the ellipse x+4y = 1. (b) Give the coordinates of the corner of the rectangle in the first quadrant, i.e. x > 0, y > 0.