Radius of a CanA can of Blue Runner Red Kidney Beans has surface ...

Question

Radius of a CanA can of Blue Runner Red Kidney Beans has surface area 371 cm^2. Its height is 12 cm. What is the radius of the circular top? Round to the nearest hun-dredth.

Accepted Answer

Hello, today we're going to be solving the following problem. So we are told that we want to find the radius of a cylindrical water bottle if the height of the bottle is 86 cm and the total surface area is 8800 cm squared. So one thing to note is that we're working with a cylindrical water bottle. And specifically, we are working with the surface area of the cylindrical water bottle. So we're going to be using the equation for the surface area of a cylinder. And that is represented as two pi R times the quantity H plus our. Now in this equation here are, represents the radius of the cylinder and H represents the height. We are told that the height of the bottle is 86 centimeters and the total surface area is 8800 centimeters squared. So we can go ahead and replace surface area or set this equation equal to 8800 And we can replace H with cm since that's going to be the height of the cylinder. So what we have is 8800 is equal to two pi R times the quantity 86 plus our. Now here, since we want to solve for the radius, we want to go ahead and sulfur are. So we want to go ahead and get this are by itself. And we're going to do that by dividing both sides of the equation by two pi. Now, one thing to note is that 8800, divided by two pi is equal to 1,400.56 with respected two decimal places. But since we want to work with integers, in this case, we're going to allow this to equal the 1400. So dividing both sides of the equation by two Pi is going to give us our times 86 plus are equal to 1400. Now, on the left hand side of this equation, we're going to distribute our into the quantity 86 plus R and that is going to give us 86 R plus R squared is equal to 1400. Now, since we want to sulfur are we want to set this equation equal to zero. So we're going to subtract both sides of the equation by 1400. And we're going to rewrite the terms with respect to the exponent. And doing this is going to give us R squared Plus 86 R -1400 is equal to zero. And what we can go ahead and do now is use the quadratic equation to sulfur are now just in case, you need to recall the quadratic equation is X is equal to negative B plus or minus the square root of B squared minus four times A times C all over two A. And this is where A B and C are the coefficients of each term of the Trine Amiel. In our current equation, the coefficient of the first term is going to be one. So A is going to equal to one. The coefficient B is the coefficient of the second term. So B is equal to 86 and C is just going to be the constant, which is going to be negative 1400. In this case. So if we want to sulfur are, we're going to set it equal to negative B which is negative 86 plus or minus the square root of B squared, which is 86 squared minus four times A which is one times C which is negative 1400 all that divided by two times A which is two times one. And now we want to go ahead and simplify in the numerator 86 squared is going to give us 7396 and negative four times one times negative 1400 is going to give us 5600. So the numerator simplifies to negative 86 plus or minus the square root of Plus 5600. And that is going to be divided by two times one, which is going to be two. Now, 7,396 plus 5600 is going to simplify to give us the square root Of 12996. So we have negative 86 plus or minus the square root of 12,996 All divided by two. And this square root itself is going to simplify to give us 114. So we are left with negative 86 plus or -114 divided by two. Now, since this is a plus and minus statement, we can go ahead and separate this into two different statements. We have R is equal to negative 86 plus divided by two. And we have R is equal to negative 86 minus 114 divided by two. On the right hand side, negative 86 minus 114 is going to give us negative 200 divided by two and negative 200 divided by two is going to give us negative 100. But one thing to note Is that any measurement, whether it be radius or behind cannot be a negative amount so are cannot equal negative 100. So now let's simplify the left hand side. Negative 86 plus 114 is going to leave us with 28. So R is equal to 28 divided by two and divided by two is going to give us 14. So that means that the radius of the cylinder is going to be cm. And with that being said, the answer to this problem is going to be a so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

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Radius of a CanA can of Blue Runner Red Kidney Beans has surface area 371 cm^2. Its height is 12 cm. What is the radius of the circular top? Round to the nearest hun-dredth.