Exam 4: Section 7: Applications of Differentiation

Find the point on the graph of the function that is closest to the point .

A sector with central angle is cut from a circle of radius 10 inches, and the edges of the sector are brought together to form a cone. Find the magnitude of such that the volume of the cone is a maximum.

Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 529 square meters.

A rectangle is bounded by the x- and y-axes and the graph of (see figure). What length and width should the rectangle have so that its area is a maximum?

Find two positive numbers whose product is 181 and whose sum is a minimum.

On a given day, the flow rate F (cars per hour) on a congested roadway is given by where v is the speed of the traffic in miles per hour. What speed will maximize the flow rate on the road? Round your answer to the nearest mile per hour.

Find the point on the graph of the function that is closest to the point . Round all numerical values in your answer to four decimal places.

Assume that the amount of money deposited in a bank is proportional to the square of the interest rate the bank pays on this money. Furthermore, the bank can reinvest this money at 36%. Find the interest rate the bank should pay to maximize profit. (Use the simple interest formula.)

The sum of the perimeters of an equilateral triangle and a square is 19. Find the dimensions of the triangle and the square that produce a minimum total area.

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 30 cubic centimeters. Find the radius, r, of the cylinder that produces the minimum surface area. Round your answer to two decimal places.

Find two positive numbers such that the sum of the first and twice the second is 56 and whose product is a maximum.

Find the length and width of a rectangle that has perimeter meters and a maximum area.

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). Find the dimensions of a Norman window of maximum area if the total perimeter is 38 feet.

A rectangular page is to contain square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

Find the length and width of a rectangle that has an area of 968 square feet and whose perimeter is a minimum.