Tanking Capacity of a Tank?

📌 CONCEPT: The problem requires us to maximize the volume of a cylindrical tank with a given surface area, which can be solved using an integral to determine the optimal dimensions of the tank.

📐 RULE / FORMULA: We can use the formula for the volume of a cylinder, V = πr^2h, where r is the radius and h is the height of the cylinder.

💡 WORKED EXAMPLE: Suppose we want to maximize the volume of a tank with a surface area of 400 m^2. Let's denote the radius as r and the height as h. We can write the volume V as an integral of the height with respect to the radius: V = ∫(πr^2) dh. To maximize V, we need to find the value of r that maximizes the integral. By solving the equation dV/dr = 0, we can find that the optimal radius is r = √(400/π) ≈ 5.65 m.

⚠️ COMMON MISTAKE: Students often forget to consider the constraint of the surface area when maximizing the volume, which can lead to incorrect results.