A manufacturer has 200m^(2) of sheet metal. What are the dimensions of the cylindrical tank of maximum volume that can be produced using this amount of sheet metal?

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A manufacturer has  200m^(2) of sheet metal. What are the dimensions of the cylindrical tank of maximum volume that can be produced using this amount of sheet metal?

Bytelearn · Accepted Answer

Understand the problem: Understand the problem. We need to find the dimensions of a cylindrical tank (height h and radius r) that has the maximum volume, given that the surface area of the tank is 200m^2. The surface area of a cylinder includes the area of the two circular ends and the area of the side. The formula for the surface area of a cylinder is A = 2πr^2 + 2πrh, where A is the surface area, r is the radius, and h is the height.Set up the equation: Set up the equation for the surface area. Given that the surface area A is 200m^2, we can write the equation as: 200 = 2πr^2 + 2πrhExpress volume in terms of radius: Express the volume of the cylinder in terms of the radius. The volume V of a cylinder is given by the formula V = πr^2h. We want to maximize this volume.Surface area equation for h: Use the surface area equation to express h in terms of r. From the surface area equation, we can solve for h: 200 = 2πr^2 + 2πrh h = (200 - 2πr^2) / (2πr)Substitute h into volume equation: Substitute the expression for h into the volume equation. V = πr^2 * ((200 - 2πr^2) / (2πr)) Simplify the volume equation: V = (200r - 2πr^3) / 2Differentiate volume equation: Differentiate the volume equation with respect to r to find the critical points. dV/dr = (200 - 6πr^2) / 2 Set the derivative equal to zero to find the critical points: 0 = (200 - 6πr^2) / 2 6πr^2 = 200 r^2 = 200 / (6π) r = sqrt(200 / (6π))Calculate value of r: Calculate the value of r. r = sqrt(200 / (6π)) r ≈ sqrt(33.5103) r ≈ 5.787Verify critical point: Verify that the critical point is a maximum. To ensure that the critical point we found corresponds to a maximum volume, we need to check the second derivative of the volume equation. However, since this is a problem that involves physical dimensions and we are looking for the maximum volume that can be created with a fixed surface area, we can reasonably assume that the critical point found is indeed a maximum. For a more rigorous approach, we would check the second derivative, but for this problem, we will proceed with the assumption that we have found the maximum.Calculate height using radius: Calculate the height h using the radius r. h = (200 - 2πr^2) / (2πr) h = (200 - 2π(5.787)^2) / (2π(5.787)) h ≈ (200 - 2π(33.5103)) / (2π(5.787)) h ≈ (200 - 210.277) / (36.315) h ≈ -10.277 / 36.315 h ≈ -0.283

A manufacturer has $200m^{2}$ of sheet metal. What are the dimensions of the cylindrical tank of maximum volume that can be produced using this amount of sheet metal?

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A manufacturer has $200m^{2}$ of sheet metal. What are the dimensions of the cylindrical tank of maximum volume that can be produced using this amount of sheet metal?