1. A farmer has 2400 ft of fencing and wants to fence off a rectangular field thatborders a straight river. He needs no fence along the river. What are thedimensions of the field that has the largest area?1. xy? + x cos y = sin y2. хеУ + ху%3D x tan y3. x In x = y2. A cylindrical can is to be made to hold 1 L of oil. Find the dimensions that will4. p = pRT is the perfect gas equation where p is pressure, p is density T istemperature and R is constant. Solve for the rate of change in pressure if therate of change in density and temperature are 12 and 5, respectively.(Disregard the units)minimize the cost of the metal to manufacture the can.3. A man launches his boat from point A on a bank of a straight river, 3 km wide,and wants to reach point B, 8 km downstream on the opposite bank, asquickly as possible. He could row his boat directly across the river to point Cand then run to B, or he could row directly to B, or he could row to some pointD between C and B and then run to B. If he can row 6 km/h and run 8 km/h,where should he land to reach B as soon as possible? (We assume that thespeed of the water is negligible compared with the speed at which the manrows.)4. Find the area of the largest rectangle that can be inscribed in a semicircle ofradius r.Implicit Differentiation5. Find two numbers whose difference is 100 and whose product is a minimum.6. Find two positive numbers whose product is 100 and whose sum is aminimum.1. Derive the following:a. xy? + x tan y = y sinb. ye* + sin xy = x tan xy7. The sum of two positive numbers is 16. What is the smallest possible value ofthe sum of their squares?8. Find the dimensions of a rectangle with perimeter 100 m whose area is aslarge as possible.9. Find the dimensions of a rectangle with area 1000 m? whose perimeter is assmall as possible.10. If 1200 cm² of material is available to make a box with a square base and anopen top, ind the largest possible volume of the box.2. Find the point on the line y = 2x + 3 that is closest to the origin.3. A farmer wants to fence in an area of 1.5 million square feet in a rectangularfield and then divide it in half with a fence parallel to one of the sides of therectangle. How can he do this so as to minimize the cost of the fence?4. Find the dimensions of the rectangle of largest area that can be inscribed in acircle of radius r.Find the Maximum/Optimum ValuesMaximum/Optimum Values and ImplicitDifferentiation

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1. A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? drical can hold 1 of oil cione