Manufacturing to Specifications. A manufacturing firm wants to...

Question

Manufacturing to Specifications. A manufacturing firm wants to package its product in a cylindrical container 3 ft high with surface area 8π ft². What should the radius of the circular top and bottom of the container be? (Hint: The surface area consists of the circular top and bottom and a rectangle that represents the side cut open vertically and unrolled.)

Accepted Answer

Welcome back. I am so glad you're here. We have this interesting problem here. Harley wants to ferment a strange fruit and decides to store his mixture inside of a barrel. That has a surface area of 1128 pi inches squared. So I'm going to draw a barrel. This is going to be the shape of a cylinder. So it's got a circle on top and a circle on the bottom. We know that the surface area of this barrel equals 1128 pi inches squared. And then for safety there should be a free space for the vapors and vigorous gasses that the process will produce. Hence this cylindrical barrel should have a height of 35 inches. So we know our height is 35 inches. We can draw that on the barrel top to bottom. The height is 35 inches. All right, what else do we know? Harley wants to design a perfect circular cover to complete the storage, calculate the radius of this cover. So, we're trying to find the radius here. That is our unknown. We're looking for our So what else do we know? How can we solve this? Will we recall from previous lessons that the surface area of a cylinder is found with the formula two pi R h plus two pi R squared. So, let's start filling in what we know to that formula? Well, we know that the whole entire thing. The surface area equals the 1128 pi inches squared. So we can set this all equal to pi. I'm going to leave off the inches for the calculation of this formula and the inches squared. We also know That we have a height of 35" so we can plug that in for h. And we're looking for are so we want the only variable down there to be our which is all that's left. Now we noticed we've got three terms here and one of them's got R squared. One of them has our and one of them has no R. This is looking like a quadratic equation. So let's put the R squared term first in order. So we'll put two pi R squared and then the R. To the first term, that's this two pi, R H. So we'll put two pi R and then R. H. That we're substituting in there. That's 35 And then that equals the 1128 pi. But let's bring that over to the left hand side so that this whole thing will be equal to zero. So this is - pi equals zero. Alright, next let's multiply this two times this. 35 here. So two pi R squared plus two times 35 is 70. So 70 pi R minus pi equals zero. Now it's just solving for R. Which is the answer that we're looking for because this is a quadratic. We need to factor. And the first thing we ask ourselves, is there anything that all three terms have in common that we can factor out? And they've all all of the regular coefficients here. These are all are there are multiples of two. So we could factor a two out. We can divide everything here by two and there's also a pie in each one of these, so we can divide everything by two pi. So the first one will leave us with just R squared plus the 70 divided by two. That's going to leave us with 35. The pie comes out and then just 35 are minus 1128 pi divided by two pi is going to leave us with 564 and that's equal to zero because zero divided by two pi is still zero. This looks much more factory ble so we can set up some parentheses to factor this, what times what gives us our squared we know it's going to be our times are for our firsts now into these last positions, we're asking ourselves what times what equals negative 564 when multiplied and adds up to a positive 35. So we've got to figure that out. Next. We can build a factor tree for 564. We know it's even So it's divisible by two. That'll be two times 282 is even. So that's two times and it will be two times 141. 141 is divisible by three, it is three times 47. So it has the factors of two times two times three times 47. Now, here are all the different ways. We can combine those factors into two things because we only have space for two numbers here that multiply to be negative 564 we no one will be positive and one will be negative. So we can take this first two times the other three multiplied together, that's two times 282. Or we can regroup it, we can take two times two which is four times three times 47 which is 141 We can group it as taking the two times the two times the three which is 12 times the 47 that's left. Or we could group it as taking A two times a three, which is six times the remaining two times 47 which is 94. So here are all of our options for those two open spaces for our lasts we know that they need one needs to be positive and one needs to be negative and they need to equal a positive 35. So the bigger numbers Because our 35 is positive, the bigger numbers should be the positive ones and the smaller ones will need to be subtracted. So which one of these? When we take the bigger number minus the smaller number equals a positive 35. That's going to be 47 -12. So we can put an R -12 and in our plus 47 and we've got it factored. So now we have two factors we can set each of them equal to zero. We have our minus 12 equals zero and R plus equals zero. We'll solve for R. Which is what we've been trying to do this whole time for the r minus 12 equals zero. We add 12 to both sides and that's our equals 12. For the r plus 47 equals zero. Will subtract 47 from both sides and we get R equals negative 47. So our radius is either going to be 12 or negative 47. So which one makes sense? Well we can't have a negative value for a radius that's just not possible. So our radius is 12. We look at our answer choices and that matches with answer choice. D well done. We'll catch you on the next one

Key Concepts

Surface Area of a Cylinder

Formulating Equations from Geometric Constraints

Solving Quadratic Equations

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