Differential Calculus

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?

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.

Water is flowing into a conical cistern at the rate of 8 \(m^3/min\). If the height of the inverted cone is 12 m and the radius of its circular opening is 6 m. How fast is the water level rising when the water is 4 m deep?

Water is pouring into a conical vessel 15 cm deep and having a radius of 3.75 cm across the top. If the rate at which the water rises is 2 cm/sec how fast is the water flowing into the conical vessel when the water is 4 cm deep?

Water is pouring into a swimming pool. After t hours, there are \(t+\sqrt{t}\) gallons in the pool. At what rate is the water pouring into the pool when t = 9 hours?

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.

A balloon is released from the ground 100 meters from an observer. The balloon rises directly upward at the rate of 4 meters per second. How fast is the balloon receding from the observer 10 seconds later?

A balloon is rising vertically over a point A on the ground at the rate of 15 ft./sec. A point B on the ground level with and 30 ft from A. When the balloon is 40 ft. from A, at what rate is its distance from B changing?