Differential Calculus

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

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?

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?

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?

A statue 3 m high is standing on a base of 4 m high. If an observer's eye is 1.5 m above the ground, how far should he stand from the base in order that the angle subtended by the statue is a maximum.

A man walks across a bridge at the rate of 5 fps as a boat passes directly beneath him at 10 fps. If the bridge is 10 feet above the boat, how fast are the man and the boat separating 1 second later?

An LRT train 6 m above the ground crosses a street at 9 m/s at the instant that a car approaching at a speed of 4 m/s is 12 m up the street. Find the rate of the LRT train and the car separating one second later.

A helicopter is rising vertically from the ground at a constant rate of 4.5 meters per second. When it is 75 m off the ground, a jeep passed beneath the helicopter traveling in a straight line at a constant rate of 80 kph. Determine how fast the distance between them changing after 1 second.

Car A moves due East at 30 kph at the same instant car B is moving S \(30^\circ\) E, with a speed of 60 kph. The distance from A to B is 30 km. Find how fast is the distance between them separating after one hour.

A car starting at 12:00 noon travels west at a speed of 30 kph. Another car starting from the same point at 2:00 P.M. travels north at 45 kph. Find how (in kph) fast the two are separating at 4:00 P.M.?