Animated Solution for Mathematics - Differentiation: A sphere of $10\text{ cm}$ radius has a uniform thickness of ice around it. Ice is melting at rate $50{\text{ cm}}^3/\text{min}$ when thickness is $5\text{ cm}$ then rate of change of thickness

Welcome, future engineer. Today, we are going to look at a problem that, at first glance, seems like a simple geometry exercise. But beneath the surface, it is a beautiful dance of calculus and physical intuition.

We are looking at a sphere of ice, melting away, and we want to know exactly how fast its thickness is changing. Let us break this down together.

The Geometry of the Shell

Imagine you are standing in a lab, looking at a solid, unmoving sphere of radius $10\text{ cm}$. This is our core, the anchor of our system.

Now, imagine we have coated this core in a uniform layer of ice. We define the thickness of this ice as $x$. The total radius of our system, from the very center to the outer edge of the ice, is therefore $r=10+x$.

To find the volume of the ice, we must isolate the ice layer. We take the volume of the entire system (the big sphere) and subtract the volume of the core. Mathematically, this is expressed as:

Notice how the core volume, $\frac{4}{3}\pi (10{)}^3$, is just a constant. This is crucial. It means that as the ice melts, the core remains unchanged, and its contribution to the rate of change will be zero.

The Calculus of Change

Now, we introduce the dynamic element: time. The problem tells us the ice is melting at a rate of $50{\text{ cm}}^3/\text{min}$. Because the volume is decreasing, we must define the rate of change as $\frac{dV}{dt}=-50$.

To find how the thickness $x$ changes with time, we need to differentiate our volume equation with respect to $t$. This is where the chain rule becomes our best friend. We differentiate the volume $V$ with respect to $x$, and then multiply by the rate at which $x$ changes with respect to $t$:

Applying the power rule and the chain rule, the derivative of $(10+x{)}^3$ becomes $3(10+x{)}^2\cdot \frac{dx}{dt}$. The constant term vanishes. Our equation simplifies beautifully:

The Final Calculation

We are almost there. We want to find the rate of change of thickness, $\frac{dx}{dt}$, at the specific moment when the thickness $x$ is $5\text{ cm}$. Let us plug in our known values:

Simplifying the term inside the parenthesis, we get $15^2$, which is $225$. Multiplying this by $4\pi$, we get $900\pi$. So, our equation becomes:

To isolate $\frac{dx}{dt}$, we divide both sides by $900\pi$:

The negative sign is not a mistake; it is a physical confirmation that the thickness is decreasing. The rate of decrease is $\frac{1}{18\pi }\text{ cm/min}$.

I know that when you first see these problems, the variables and the rates can feel overwhelming. But look at what we just did. We took a physical process—melting ice—and translated it into the language of mathematics.

We visualized the geometry, applied the chain rule, and arrived at a precise, elegant solution. Keep practicing this translation, and you will find that physics is not just about formulas; it is about telling the story of how the world changes. You have got this!