A farmer with 2000 meters of fencing wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed?

A farmer wants to fence in a rectangular plot of adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. If the fencing costs $20 per linear foot to install and the farmer is not willing to spend more than$5000, find the dimensions for the plot that would enclose the most area.

A farmer with 2000 meters of fencing wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed?

A farmer wants to fence an area of 24 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. What should the lengths of the sides of the rectangular field be so as to minimize the cost of the fence?

If 10,800 cm^2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

A small plane flies 40.0 km in a direction 60° north of east and then flies 30.0 km in a direction 15° north of east. Use the analytical method to find the total distance the plane covers from the starting point, and the geographic direction of its displacement vector. What is its displacement vector?

An observer stands at a point $P$, one unit away from a track. Two runners start at the point $S$ in the figure and run along the track. One runner runs three times as fast as the other. Find the maximum value of the observer’s angle of sight $\theta$ between the runners. [Hint: Maximize $\tan \theta$.]

A farmer wants to fence a rectangular area by using the wall of a barn as one side of the rectangle and then enclosing the other three sides with $160$ feet of fence. Find the dimensions of the rectangle that give the maximum area inside.

Use Newton's method to find all solutions of the equation correct to six decimal places.

$$\ln x=\frac{1}{x-3}$$

A homeowner has $200$ feet of fence to enclose an area for a pet.

(a) If the area is given by $A(x)=x(100-x)$, what dimension maximize the area inside the fence?

(b) What is the maximum area?

($c$) Determine the domain and range of $A$ in this application.

Use the method of Lagrange multipliers to find the dimensions of the rectangle of greatest area that can be inscribed in the ellipse

$$x ^ { 2 } / 16 + y ^ { 2 } / 9 = 1$$

with sides parallel to the coordinate axes.

If the farmer in previous exercise wants to enclose 8000 square feet of land, what dimensions will minimize the cost of the fence?

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, as quickly as possible. He could row his boat directly across the river to point C and then run to B, or could row directly to B, or he could row to some point D 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 the speed of the water is negligible compared with the speed at which the man rows.)

A model used for the yield of an agricultural crop as a function of the nitrogen level N in the soil (measured in appropriate units) is y = kN / (1 + N^2) where k is a positive constant. What nitrogen level gives the best yield?