Cylinder Volume Change: Finding Radius Rate
Hey guys! Let's dive into a super interesting math problem today involving cylinders and how their volume changes when both height and radius are in motion. This is a classic calculus problem that combines geometry with rates of change, and it’s something you might encounter in your studies or even in real-world applications. So, buckle up, and let's get started!
Understanding the Problem
In this problem, we're dealing with a cylinder whose dimensions are changing over time. Specifically, the height of the cylinder is increasing at a constant rate of 8 centimeters per minute, and the volume is increasing at a constant rate of 540 cubic centimeters per minute. Our main goal here is to figure out how the radius of the cylinder is changing at a particular instant. To solve this, we'll need to use our knowledge of the formula for the volume of a cylinder, as well as some calculus magic—specifically, related rates.
Key Concepts: Related Rates
Before we jump into the nitty-gritty, let's quickly recap what related rates are all about. Imagine you have several variables that are all changing with respect to time, and these variables are related by an equation. Related rates problems ask you to find the rate at which one variable is changing, given the rates of change of the other variables. The secret sauce? Implicit differentiation. We'll differentiate the equation with respect to time (usually denoted as t), and then we'll plug in the known rates to solve for the unknown rate. It's like being a detective, but with math!
Setting Up the Problem
Okay, let's break down our specific problem step-by-step. This structured approach will help us keep things organized and clear. Trust me, in calculus, organization is your best friend.
1. Identify the Variables
First things first, we need to identify all the variables involved and what they represent:
- V: Volume of the cylinder (in cubic centimeters)
- r: Radius of the cylinder (in centimeters)
- h: Height of the cylinder (in centimeters)
- t: Time (in minutes)
2. Write Down the Given Rates
Next, let's jot down the rates of change that we know from the problem statement. Remember, rates of change are just derivatives with respect to time:
- dh/dt = 8 cm/min (the height is increasing at 8 cm/min)
- dV/dt = 540 cm³/min (the volume is increasing at 540 cm³/min)
3. State What We Need to Find
It's crucial to know exactly what we're trying to solve for. In this case, we want to find dr/dt, which is the rate of change of the radius with respect to time. This is the big question mark we're aiming to answer.
4. The Volume Formula
The backbone of our solution is the formula for the volume of a cylinder, which you probably remember from geometry:
V = πr²h
This formula relates our variables V, r, and h, and it’s the equation we'll differentiate to connect their rates of change.
Solving the Related Rates Problem
Now comes the fun part—the calculus! We're going to use implicit differentiation to find the relationship between the rates, and then we'll plug in the values we know to solve for the unknown rate.
1. Differentiate with Respect to Time
We need to differentiate both sides of the volume formula V = πr²h with respect to time (t). This is where the chain rule and product rule come into play. Remember, r and h are both functions of time, so we need to treat them as such when differentiating.
d/dt (V) = d/dt (πr²h)
Applying the differentiation rules, we get:
dV/dt = π (2r (dr/dt) h + r² (dh/dt))
Notice how the chain rule pops up when differentiating r² (giving us 2r dr/dt) and the product rule is used because we have the product of r² and h.
2. Plug in the Known Values
Alright, we have our equation relating the rates. Now it’s time to plug in the values we know. The problem statement gives us dV/dt = 540 cm³/min and dh/dt = 8 cm/min. We also need to know the values of r and h at the specific instant we're interested in. Let's say, for the sake of example, that at a certain moment, r = 6 cm and h = 15 cm. (These values would typically be given in the problem, or you might need to find them using other information.)
Plugging in these values, our equation becomes:
540 = π (2 * 6 * (dr/dt) * 15 + 6² * 8)
3. Solve for dr/dt
Now we just need to do some algebra to isolate dr/dt. Let's simplify the equation:
540 = π (180 * dr/dt + 36 * 8) 540 = π (180 * dr/dt + 288)
Divide both sides by π:
540/Ï€ = 180 * dr/dt + 288
Subtract 288 from both sides:
540/Ï€ - 288 = 180 * dr/dt
Divide by 180:
dr/dt = (540/Ï€ - 288) / 180
Calculate the value:
dr/dt ≈ (171.89 - 288) / 180 dr/dt ≈ -116.11 / 180 dr/dt ≈ -0.645 cm/min
So, at this particular instant, the radius is decreasing at a rate of approximately 0.645 centimeters per minute. The negative sign tells us that the radius is shrinking.
Importance of Units
Always, always, always include units in your final answer! It's not just a formality; it helps you make sure your answer makes sense. In this case, dr/dt is in centimeters per minute (cm/min), which is a rate of change for a length (radius) over time. If we had ended up with cubic centimeters per minute, we'd know something went wrong!
Real-World Applications
You might be wondering,