I have two questions of optimization that seem almost impossible. If someone cou

Question

I have two questions of optimization that seem almost impossible. If someone could solve them for me and show me how they do them this is a review paper I was given for a test coming up. 1.A manufacturer believes that when x units of widgets are produced each month, the total cost (in dollars) will be C(x)=1/8x^2 + 4x + 200. Additionally, all widgets can be sold at a price of P(x)=(49-x) dollars per widget. a. determine the price that corresponds to the maximum profit. b. determine the level of production where the average cost is minimized 2. Two sides of a triangle are 4 inches long. What should the angle betweent he sides be to make the area of the triangle as large as possible

Explanation / Answer

1. Write an expression for the profit. Find the maximum by differentiating, setting equal to 0 and solving for x. a. profit(x) = P(x)*x - C(x) ...(P(x) is multiplied by x because it is a price per widget) = (49 - x)*x - 1/8*x^2 - 4x - 200 = -9/8*x^2 + 45x -200 d/dx(profit) = -18/8*x + 45 = 0 ---> x = 20 Price per widget if 20 widgets are made = 49 - 20 = $29 / widget. b. find minimized average cost by differentiating and setting equal to 0, solving for x. C(x) is TOTAL coast, divide by number of widgets, x, to get average cost per widget. C(x)/x = 1/8*x + 4 + 200/x d/dx = 1/8 - 200/x^2 = 0 200/x^2 = 1/8 x^2 = 200*8 = 1600 x = 40 2. The area for a triangle is .5*b*h. You're given b=4. to solve for h we use h = 4*sin(180-theta)=4*sin(theta). Draw it out so you can visualize why these equations are valid. A = .5*4*4*sin(theta) = 8*sin(theta) dA/dtheta = 8*cos(theta) = 0 cos(theta) = 0 when theta is 90 degrees

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