Contextual Applications of Differentiation

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 $0.02$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?

An ant is moving on the coordinate plane. When the ant is at point (x, y), the angle of the ant is defined as $\arctan\left(\frac{y}{x}\right)$. 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 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 spherical balloon is being inflated such that its volume increases at a rate of $12\pi \text{ cubic inches per minute}$. What is the rate of increase in the radius when it's equal to $3 \text{ inches}$?

A ladder $10$ ft long rests against a vertical wall; if the bottom starts slipping away from the wall at a rate of $1$ ft/sec, how fast is the top sliding down when the bottom is $6$ ft from the wall?

A spherical balloon is being pumped with air at a rate of $16\pi \text{ cm}^3/ ext{s}$, which causes the balloon’s radius to increase at an instantaneous rate of $0.04 \text{ cm/s}$. What is the radius of the balloon at this moment?

A car is driving along a straight road at a speed of 72 km/h. A police officer is standing 60 meters away from the road. How fast is the distance between the car and the police officer changing when the car is 80 meters from the officer?