A customer has asked you to design an open-top rectangular sta...

Question

A customer has asked you to design an open-top rectangular stainless steel vat. It is to have a square base and a volume of 32 ft³ , to be welded from quarter-inch plate, and to weigh no more than necessary. What dimensions do you recommend?

Accepted Answer

Welcome back, everyone. A chemical company needs a new open top cylindrical tank to store a chemical solution. The tank must have a volume of 50 m cubed. To minimize the material cost, the surface area of the tank, including the base but not the top, should be minimized. What should be the radius and height of the tank in meters. So now let's break it down. First of all, we know that our volume is constant. It is equal to 50 m cubed. There's no necessity to write any units for now. We're just going to set B equals 50, and we know that it is a cylinder. So what we're going to do is simply identify the area of the base, which is by our square that is circular, and then we are multiplying by height H, right? And another formula that we want to use is the surface area, but we have to understand that it is an open top cylinder, right? So this means that we're adding one base with an area of pi R squared, and then we are going to add two pi RH which essentially corresponds. To the lateral area, right? So this is our formula and we can also factor out IR as a common factor which leaves us with R + 2 H and Cs. Now we have our constraint and our objective function. So let's call 1 as C. That's our constraint function, and we have our objective function O. So typically what we want to do is use our constraint and express one of the variables in terms of the other so we can express height. In terms of radius, we are going to take 50 on the left hand side of the equation and divide it by pi or square. So now what we're going to do is simply use this and substitute into the surface area formula so that we only have one. Variable this gives us. S surface area equals pi R multiplied by R plus 2 H. So this gives us 2 multiplied by 50, which is 100 divided by pi R squad. Well done, so now we have our objective function expressed in terms of R only. And what we're going to do is simply find its derivative. So to avoid the product rule, we can just distribute the terms, and we're going to get pi R squared the gain. Plusi are divided by pi 2 this gives us 100 divided by r. And now let's identify the derivative of S with respect to R. Now the derivative of pi R squared is 2 pi R based on the power rule, and now to differentiate 100 divided by R we can just rewrite it as. 100R raises the power of -1. So we bring down -1, which gives us a negative sign, and then we get 100 to the power of -2, which is simply 100 divided by R squad, right? And now because we want to optimize the surface area, we're going to set it equal to 0. The derivative does not exist when R is equal to 0, but in any case, R must be greater than 0, right? Based on the context of the problem, we cannot have a zero radius for the cylinder to exist. So if we're setting it equal to 0, we need to find the common denominator which is R2d. So we're multiplying 2 R by R2 which gives us 2R cubed minus 100 in the numerator, and this is equal to 0. So we essentially need to set our numerator equal to 0. This means that 2 pi cubed is equal to 100. And therefore our cubed is equal to 100 divided by 2 pi and now we can take the cubic root of both sides to get r on its own right, so we're taking the cubic root of that. And we can also simplify the fraction because both sides are divisible by 2, so we get 50 divided by pi. And this is how we have found the radius, right, which potentially minimizes the surface area, but we need to check if this radius indeed corresponds to a local minimum. So we're going to apply the second derivative test and we're going to identify as prime with respect to R. So the derivative of 2 piR is just 2 pi. And now once again, we have 100 divided by R squad, which is simply 100 R race to the power of -2, right? We also have a negative sign in front. So if we want to find a derivative, we bring down -2, which gives us plus 2. Multiplied by 100 are raised to the power of -3 based on the power rule. So now we can rewrite it in a more readable form. This gives us 2 by. Plus 200 divided by R cubed. And now what we want to do is simply evaluate the second derivative at the critical point, so cubic root. Of 50 divided by pi. So let's substitute. We're going to get 2 pie. Plus 200 divided by our cubed. So this means we simply get rid of the radical sign and we are dividing by 50 divided by pi. Let's perform the calculations. We get 2 pi. Plus 200 I. Divided by 50. Which is equal to 4 + 2 and that's 6. Now we can sell that this is greater than 0, meaning our function is concave up. At the critical point of it, and if it's concave up, we have a local minimum, so we have verified that the given r value indeed corresponds to the minimum value. So the remaining part that we want to solve is height. So height is equal to 50 divided by pi R squad. So we take 50 divided by pi multiplied by R2. So To perform the calculations, we're going to substitute R. And we can say that R is simply 50 divided by pi raises the power of 1/3, right? And now we are squaring it. So if we square it, we have to multiply the two exponents, so to multiply it by 1/3 gives us 2/3, right? And now we are going to simplify, so we have 50 divided by. Pi, which is multiplied by 50, raised to the power of 2/3 and divided by pirates to the power of 2/3. So we have distributed the exponent and now let's keep simplifying. So first of all, we can write it as 50 divided by. 5th T raises the power of 2/3 by, and then we are once again dividing by pi raise to the power of 2/3. So if we have a double division, we can just write i raise to the power of 2/3 in the numerator. And from here we can apply the properties of exponents. So first of all, we have 50, raise the power of 1 divided by 50, raise the power of 2/3. So 1 minus 2/3 gives us 1/3. And now for the pi terms, we have i raise the power of 2/3 divided by pi raises the power of 1. So 2/3 minus 1 gives us negative 1/3. And we can see that this is the same as cubic root. Of 50 divided by pi because we have 50. Raise the power of 1/3, which is cubic root of 50, and pi raises the power of -1/3 is cubic root of 1 divided by pi we just combine them under the same radical, and we can see that R and H, they essentially correspond to the same values. We can conclude that R equals H equals cubic root. Of 50 divided by pi and our units would be meters, right? Because volume was given in meters cubed, and that will be our final answer to this problem. Thank you for watching.

A customer has asked you to design an open-top rectangular stainless steel vat. It is to have a square base and a volume of 32 ft³ , to be welded from quarter-inch plate, and to weigh no more than necessary. What dimensions do you recommend?

Key Concepts

Volume of a Rectangular Prism

Optimization

Surface Area Calculation

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