Suppose \(x^2 + 4y^2 = 40\), where \(x\) and \(y\) are functions of \(t\). If \(\dfrac{dx}{dt} = 1.5\), find \(\dfrac{dy}{dt}\) when \(x = 2\) and \(y = 3\).
The radius of a circle is increasing at the rate of 2 inches per sec. At what rate is the area increasing when the radius is 5 inches?
Each side of a square is increasing at a rate of 3 inches per min. At what rate is the area of the square increasing when the side is 7 inches?
In calm waters, oil spilling from a ruptured tanker spreads in all directions. Assuming the polluted area is circular, determine how fast the area is increasing when the radius of the circle is 60 feet and is increasing at the rate of \(\tfrac{1}{2}\) feet per sec.
The radius of a sphere is increasing at a rate of 8 mm/s. How fast is the volume increasing when the diameter is 40 mm?
A cylindrical tank with radius 6 m is being filled with water at a rate of 5 cubic m/min. How fast is the height of the water increasing?
A ladder 17 feet long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1.2 ft/sec, how fast is the angle between the ladder and the ground changing when the bottom is 8 ft from the wall?
Two cars start moving from the same point. One travels south at 60 mph and the other travels west at 25 mph. At what rate is the distance between the cars increasing two hours later?
A street light is mounted at the top of a 15 ft tall pole. A man 6 ft tall walks away from the pole with a speed of 5 ft/sec along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole?
A girl flies a kite at a height of 300 ft, the wind carrying the kite horizontally away from her at a rate of 25 ft/sec. How fast must she let out the string when the kite is 500 ft away from her? Assume the string is taut and forms a straight line.
Person A starts walking north at 3 ft/s from a point P. Five minutes later, Person B starts walking south at 4 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 10 minutes after Person B starts walking?
A balloon is rising at a constant speed of 5 ft/sec. A boy is cycling along a straight road at a speed of 15 ft/sec. When he passes under the balloon, it is 45 ft above him. How fast is the distance between the boy and the balloon increasing 3 sec later?
The altitude of a triangle is increasing at a rate of 1.5 cm/min while the area is increasing at a rate of 5 cm²/min. At what rate is the base of the triangle changing when the altitude is 9 cm and the area is 81 cm²?