Differential Calculus

Find the minimum distance from the point \((4,2)\) to the parabola \(y^2=8x\).

The sum of two positive numbers is 50. What are the numbers if their product is to be the largest possible.

A triangle has variable sides x,y,z subject to the constraint such that the perimeter is fixed to 18 cm. What is the maximum possible area for the triangle?

A farmer has enough money to build only 100 meters of fence. What are the dimensions of the field he can enclose the maximum area?

Find the minimum amount of tin sheet that can be made into a closed cylinder having a volume of 108 cu. inches in square inches.

A box is to be constructed from a piece of zinc 20 sq.in by cutting equal squares from each corner and turning up the zinc to form the side. What is the volume of the largest box that can be so constructed?

A poster is to contain 300 (cm square) of printed matter with margins of 10 cm at the top and bottom and 5 cm at each side. Find the overall dimensions if the total area of the poster is minimum.

A norman window is in the shape of a rectangle surmounted by a semi-circle. What is the ratio of the width of the rectangle to the total height so that it will yield a window admitting the most light for a given perimeter?

Determine the diameter of a closed cylindrical tank having a volume of 11.3 cu. m to obtain minimum surface area.

The cost of fuel in running a locomotive is proportional to the square of the speed and is $ 25 per hour for a speed of 25 miles per hour. Other costs amount to $ 100 per hour, regardless of the speed. What is the speed which will make the cost per mile a minimum?

The cost C of a product is a function of the quantity x of the product: \(C(x)=x^2-4000x+50\) Find the quantity for which the cost is minimum.

An open top rectangular tank with square bases is to have a volume of 10 cu. m. The materials for its bottom is to cost P 15 per square meter and that for the sides, P 6 per square meter. Find the most economical dimensions for the tank.

What is the maximum profit when the profit-versus-production function is as given below? P is profit and x is unit of production. \(P = 200{,}000 - x - (\frac{1.1}{x+1})^8\)

A boatman is at A which is 4.5 km from the nearest point B on a straight shore BM. He wishes to reach in minimum time a point C situated on the shore 9 km from B. How far from C should he land if he can row at the rate of 6 kph and can walk at the rate of 7.5 kph?

A fencing is limited to 20 ft length. What is the maximum rectangular area that can be fenced in using two perpendicular corner sides of an existing wall?

The cost per hour of running a motor boat is proportional to the cube of the speed. At what speed will the boat run against a current of 8 km/hr in order to go a given distance most economically?

Given a cone of diameter x and altitude of h. What percent is the volume of the largest cylinder which can be inscribed in the cone to the volume of the cone?

At any distance x from the source of light, the intensity of illumination varies directly as the intensity of the source and inversely as the square of x. Suppose that there is a light at A, and another at B, the one at B having an intensity 8 times that of A. The distance AB is 4 m. At what point from A on the line AB will the intensity of illumination be least?

A wall "h" meters high is 2 m away from the building. The shortest ladder that can reach the building with one end resting on the ground outside the wall is 6 m. How high is the wall in meters?

The coordinates (x,y) in feet of a moving particle P are given by \(x=\cos t-1\) and \(y=2 \sin t+1\), where t is the time in seconds. At what extreme rates in fps is P moving along the curve?