Minimum painting surface A metal cistern in the shape of a rig...

Question

Minimum painting surface A metal cistern in the shape of a right circular cylinder with volume V = 50 m³ needs to be painted each year to reduce corrosion. The paint is applied only to surfaces exposed to the elements (the outside cylinder wall and the circular top). Find the dimensions r and h of the cylinder that minimize the area of the painted surfaces.

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says an environmental research center is designing a cylindrical rainwater collector with a volume of 100 cubic meters to study rainfall patterns in arid regions. The collector will be opened at the top, but its sides and bottom surfaces need to be coated with a special material to prevent evaporation. Find the dimensions of the collector the radius R and height H. That would minimize the surface area to be coded. We're given 4 possible choices as our answers. For choice A, we have R is equal to the quantity of 100 divided by pi in quantity rates to the quantity of 1 divided by 3 in quantity meters, and H is equal to the quantity of 100 divided by pi in quantity, rates to the quantity of 1 divided by 3 in quantity meters. For choice B, we have Rs equal to the quantity of 50 divided by pi in quantity raised to the quantity of 1 divided by 3 in quantity meters, and H is equal to 100 divided by the quantity of pi multiplied by the quantity of 50 divided by pi in quantity raised to the quantity of 32 divided by 3. In quantity meters. For C, we have R is equal to the quantity of 10 divided by pi in quantity raised to the quantity of 1 divided by 3 in quantity meters, and H is equal to 100 divided by the quantity of pi multiplied by the quantity of 10 divided by pi in quantity raised to the quantity of 2 divided by 3 in quantity meters. And for choice D we have R is equal to 100 divided by the quantity of pi multiplied by the quantity of 5 divided by pi in quantity raised to the quantity of 2 divided by 3 in quantity meters, and H is equal to the quantity of 5 divided by pi in quantity raised to the quantity of 1 divided by 3 in quantity meters. Now we're asked to find the dimensions of this collector that would minimize the surface area to be coded. So recall your formula for the surface area of a cylinder, we will have A, which is our area, which is going to be equal to pi R squad. Plus 2 pi are multiplied by H. So here the pier squared is just going to be the bottom surface, and the two piR H is going to be the outer cylindrical surface. Now, we actually want to minimize the surface area. So we actually want to take a derivative of our area. However, we have two variables here. We have R and H. What we want to, what we really want to do is write one of these variables in terms of the other. So here we need to recall the volume of a cylinder. Recall that the volume of a cylinder, V. It's going to be equal to. Pi R2 multiplied by H. We're told that our volume is a constant, so it's 100 cubic meters. That means that this is going to be equal to 100. And so we can actually solve this equation here for H by dividing both sides of the equation by pi R2. So we'll have H is equal to 100 divided by the quantity of pi R squared. So now we can substitute in for 8 this quantity and have our area written in terms of just one variable. In doing so, that means that our area A going to be equal to IR squad. Plus 2R Multiplied by the quantity of 100 divided by the quantity of pi or squared in quantity. So now we can simplify this expression, so this is going to be equal to. Pi R squared. Plus 200 Divided by R. So now we have our area here written in terms of just a single variable. We can take the derivative of our area with respect to R and set it equal to 0 and solve for R, and this will give us the radius that minimizes the surface area. Then we then can use that radius to calculate our height. So, we're gonna take the derivative of A. With respect to R, and that's gonna be equal to the derivative with respect to R. Of the quantity of pi R2 plus 200 divided by R. So taking the derivative, this is going to be equal to, for the first term. When I take the derivative with respect to R of pi R2, I get 2 pi R. Then we'll have less for the derivative of a second term, so we'll have the 200 divided by R. We'll have the derivative with respect to R of that quantity. And so taking that derivative, we're going to get the quantity of minus 200 divided by R squared in quantity. And now we want to do is set this equal to 0. So we will have 2 R minus 200 divided by R squared. is equal to 0. So now we need to just solve this for R. So the first step is to add 200 divided by R2 to both sides of this equation. So we'll have 2 r is equal to 200 divided by R2. Now we'll just multiply both sides of this equation by R2d and divide both sides by the quantity of 2 pi. Doing so, we'll get R cubed. is equal to 200 divided by 2 pi. And now we just need to raise everything to the 1/3 power. The sul for R, so we'll have R is equal to the quantity of 100 divided by pi in quantity raised to the quantity of 1 divided by 3 in quantity. So this here is the value of R that we're looking for. So now we just need to use it to calculate the value for H. To recall that age. is equal to. 100 Divided by the quantity of IR squad. So substituting in our value for R, this is going to be equal to 100. Divided by the quantity of pi multiplied by the quantity of 100 divided by pi in quantity, raised to the quantity of 2 divided by 3. In quantity And so when we simplify this fraction, this is going to be equal to the quantity of 100. Divided by i in quantity raised to the 1 divided by 3 power. So this here is Our value for H that we were looking for. So now that we have both values calculated, we can compare that to our answer choices, and we see that the correct answer choice corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Volume of a Cylinder

Surface Area of a Cylinder

Optimization Techniques

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