Contextual Applications of Differentiation
An ant is moving on the coordinate plane. When the ant is at point (x, y), the angle of the ant is defined as . If the ant is currently located at the point (5, 5), its x-coordinate is changing at a rate of -2 units/second, and its y-coordinate is changing at the rate of 4 units/secon...
A rectangular gate on a dam consists of two parts—upper triangle and lower rectangle—with water pressure causing forces on both; how fast does force change on the rectangular part if water rises at a constant rate of m/min?
If the rate of change of the radius (r) of a circle with respect to time (t) is constant, how does the area (A) change over time?
The length of a rectangular prism is increasing at a rate of 6 in/s, the width is increasing at a rate of 4 in/s, and the height is decreasing at a rate of 2 in/s. If the length is 8 in, the width is 6 in, and the height is 4 in, what is the rate of change of the volume with respect to time?
A cubical tank with a side length of 5 meters, with one of its faces on the ground, is being filled with water at a rate of 20 m^3/hr. How fast is the water level rising?
A marathon runner is jogging around a circular track with a radius of 60 meters at a constant speed of 4 m/s, while her running partner stands still at the starting point. Exactly 10π seconds after she leaves the starting point, what is the rate of change in the distance between the runner and her partner?
A drop of water in space is currently in the shape of a cylinder with radius 1 mm and height 2 mm. A piston pushes the circular faces of the drop closer together, decreasing the height at a rate of 1 mm/s. If the water stays in the shape of a cylinder and its volume does not change, at what rate is the radius of the dr...
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The radius of a circular pool is decreasing at a rate of 0.25 m/min. At what rate is the area of the pool changing when the radius is 4 m?
A water tank in the shape of a cone is being filled with water at a rate of 20 cubic meters per minute. If the height of the tank is 10 meters and the radius is 4 meters, how fast is the water level rising when the water is 5 meters deep?
The length of a rectangle is increasing at a rate of 2 cm/s, the width is increasing at a rate of 3 cm/s, and the area is increasing at a rate of 56 cm^2/s. If the length of the rectangle is currently 12 cm/s, what must be the width of the rectangle?