Solve optimization problems

A man needs to travel from point A to point B , where point B is away, vertically opposite through the river, while the horizontal distance of point A to B is . The man needs to reach point B in minimum time. His speed on the ground is and in the water his speed is . Find out the horizontal distance from A from where the man needs to enter the water in order to reach the point B in minimum time.

Find a point on the function closest to the point .

A farmer wants to build a fence enclosing a rectangular area of . He wants to use the least amount of fencing material. Determine the dimensions of the fence that will have the least perimeter. 

A student needs to create a cylinder in one of his projects. The cylinder must have a volume of  . Find the radius of the cylinder that will minimize the amount of material that will be used for the cylinder. 

It takes of paint to cover of rough flooring. If a rectangular area is required to be painted and the perimeter of the area should be , how much paint will be needed if the rectangular area is maximized? 

A cylindrical vessel will be constructed for a space mission. If the surface area of the vessel should be , what is the maximum volume that the vessel can hold?

A by   rectangular sheet of paper was used to create a box with a lid. The sides of the box are folded upward and square with side are cut from the corners. What should be the dimensions of the box to have maximum volume possible for the box?

A   by   rectangular sheet of paper was used to create a box with a lid. The sides of the box are folded upward and square with side are cut from the corners. What is the maximum volume possible for the box?

A wooden rectangular frame for molding cement will be made. The base of the frame needs to have an area of . If the woodworker wants to minimize the material used for the perimeter of the base, what should be the perimeter of the base?

A rope that is long will be used to create a rectangle. What should be the dimension of the rectangle to obtain the maximum area?

and are points on the y-axis. Find the x-coordinate of point on the x-axis such that the angle has the maximum possible value.

Suppose Ashley is making a rectangle shape using a single rope, which has a length of , what is the maximum area of the rectangle that Ashley can make up from the given rope?

What is the smallest possible perimeter of a rectangle with area ?

A laptop screen needs to be at the correct angle with respect to horizontal for the viewer to have the best experience. The user experience with respect to the angle of the screen with respect to the horizontal is given by . Find out the angle at which the experience is best for the user if angle with horizontal is supposed to be more than and less than radians.

A Norman window is to be made out of the Plastic beam, whose available length is . Find out the dimensions of the Norman window such that when the window is installed into the wall, maximum Air circulation is possible. A Norman window is a window wherein one side of the rectangle is replaced by a semicircle.