Air fills a spherical balloon. Its radius is $r$ centimeters and volume is $V$ cubic centimeters. Which relationship should be differentiated with respect to time?

A circular oil spill has radius $r$ meters and area $A$ square meters, both changing with time. Which equation relates the quantities?

Water occupies a cylinder with water-surface radius $r$ centimeters, depth $h$ centimeters, and volume $V$ cubic centimeters. Which relationship should be used?

A conical tank contains water with surface radius $r$ centimeters, depth $h$ centimeters, and volume $V$ cubic centimeters. Which equation relates these quantities?

The side length $s$ centimeters of an expanding square determines its area $A$ square centimeters. Which relationship should be differentiated with respect to time?

A rectangular sheet has changing length $\ell$ centimeters, width $w$ centimeters, and area $A$ square centimeters. Which equation relates the quantities?

A rectangular box has changing length $\ell$ meters, width $w$ meters, height $h$ meters, and volume $V$ cubic meters. Which equation should be differentiated with respect to time?

A ladder of fixed length 25 centimeters has foot distance $x$ from a wall and top height $y$ on the wall, so $x^2+y^2=625$. Differentiate with respect to $t$.

A spherical balloon satisfies $V=\frac{4}{3}\pi r^3$, where both $V$ and $r$ change with time. Differentiate with respect to $t$.

A circular region satisfies $A=\pi r^2$, where its area and radius change with time. Differentiate with respect to $t$.

A cylinder has fixed height $h$ centimeters, changing radius $r$ centimeters, and volume $V=\pi r^{2}h$. Differentiate with respect to $t$.

A rectangle satisfies $A=\ell w$, where its length $\ell$ and width $w$ both change with time. Differentiate with respect to $t$.

Water in a cone satisfies $V=\frac{1}{3}\pi r^{2}h$, where $r$ and $h$ both change with time. Differentiate with respect to $t$.

A drone has horizontal distance $x$ meters, altitude $y$ meters, and straight-line distance $z$ meters from a station, with $z^2=x^2+y^2$. Differentiate with respect to $t$.

Water drains from a tank at 9 liters per minute. Volume is $V$ liters and water depth is $h$ centimeters. The question asks how fast the depth falls. What are the given and wanted rates?

Air enters a spherical balloon, and its radius $r$ grows at 0.4 centimeters per second. The question asks how fast volume $V$ changes. What are the given and wanted rates?

A rectangle has length $\ell$ increasing at 3 centimeters per second and width $w$ decreasing at 1 centimeter per second. The question asks for the area rate. Which symbols state this?

A person walks away from a lamp. Let $x$ be the person's distance from the lamp and $s$ be the shadow length. If the question asks how fast the shadow length changes, what are the given and wanted rates?

Water rises in a conical tank at 2 centimeters per second. Let $h$ be water depth and $V$ be water volume. The question asks for the volume rate. Which symbols state this?

At one instant a spherical balloon has radius $r=5$ centimeters and $V=\frac{4}{3}\pi r^3$. Which setup correctly finds its volume rate?

A cylinder has $V=\pi r^{2}h$. At an instant, $r=3$ centimeters and $h=10$ centimeters, and both dimensions are changing. Which step is correct?

A fixed 25-centimeter ladder has $x^2+y^2=625$. At the instant considered, $x=7$ and $y=24$. Which setup correctly preserves the related rates?

A conical tank has top radius 6 centimeters and height 18 centimeters, so the water's surface radius $r$ and depth $h$ satisfy $\frac{r}{h}=\frac{6}{18}$. If only $\frac{dh}{dt}$ is known, which volume formula should be differentiated?

A conical tank has top radius 4 centimeters and top height 10 centimeters. Water-surface radius $r$ and depth $h$ always satisfy $\frac{r}{h}=\frac{4}{10}$. Which equation eliminates $r$ before differentiating?