Differential Calculus

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?

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.?

Two railroad tracks are perpendicular to each other. At 12:00 P.M. there is a train at each track approaching the crossing at 50 kph, one being 100 km and the other 150 km away from the crossing. How fast in kph is the distance between the two trains changing at 4:00 P.M.?

Water is running into a hemispherical bowl having a radius of 10 cm at a constant rate of 3 \(cm^3/min\). When the water is x cm. deep, the water level is rising at the rate of 0.0149 cm/min. What is the value of x?

What is the allowable error in measuring the edge of the cube that is intended to hold 8 cu. m., if the error of the computed volume is not to exceed 0.03 cu. m?

A standard cell has an emf "E" of 1.2 volts. If the resistance "R" of the circuit is increasing at the rate of 0.03 ohm/sec, at what rate is the current "I" changing at the instant when the resistance is 6 ohms? Assume Ohm's law E = IR.

Sand is pouring to form a conical pile such that its altitude is always twice its radius. If the volume of the conical pile is increasing at the rate of 25π cu.ft/min, how fast is the radius increasing when the radius is 5 feet?

A wall 8 feet high is 3.375 feet from a house. Find the shortest ladder that will reach from the ground to the house when leaning over the wall.

A coat of paint of thickness 0.01 inch is applied to the faces of a cube whose edge is 10 inches, thereby producing a slightly larger cube. Estimate the number of cubic inches of paint used.

The tangent line to the function h(x) at (6, −1) intercepts the y-axis at y = 4. Find \(h'(6)\).

A particle has a position vector \((2\cos 2t,\ 1 + 3\sin t)\). What is the speed of the particle at time t = π/4?