Please help with this problem regarding cylinders!

[janemercado]

 

[MarkFL]

Let's present the problem so it is more easily read:

10.) The figure below shows a right prism, the base of which is a quarter of a circle with center $C$. If the area of each base of the prism is $12.5\pi$ and the volume of the solid is $300\pi$, what is the distance from point $A$ to point $B$?



Can you show what you've done so far, and where you're stuck?

 

[janemercado]

Yes, of course.The first thing I did is use the value for area of the base to find the radius, and I did this by dividing the area of a circle formula by 4 (b/c you only have a quarter of a circle). The value of the radius turned out to be the square root of 50, and with this value, I found the height of the cylinder by using the volume given, and by once again dividing the volume of a cylinder formula by 4. The height came out to be 24 units. The 24 units would be the length of the cylinder, and also the length of the rectangle the cylinder made when rolled out. Now with these pieces of information, I calculated the circumference of the circle (which would represent the with of this rectangle. This came out to a messy fraction and the result of these calculations is shown in the picture I made. Lastly, I used the pythagorean theorem to find the hypotenuse of the rectangle, as this would be the distance between the two points. However, when I checked with my teacher, he said my answer was incorrect.

 

[janemercado]

I am unsure of where I made a mistake, as I was pretty confident about my calculations. Is there a way I could guide my work from what I have already to achieve the correct answer?

 

[MarkFL]

Okay, let me check the value you found for the radius of the bases:

[MATH]\frac{25}{2}\pi=\frac{1}{4}\pi r^2[/MATH]
[MATH]50=r^2[/MATH]
[MATH]r=\sqrt{50}=5\sqrt{2}\quad\checkmark[/MATH]
And next the height:

[MATH]\frac{25}{2}\pi h=300\pi[/MATH]
[MATH]h=24\quad\checkmark[/MATH]
Then what I would do (which appear to be what you did) is consider that points $A$ and $B$ are opposing vertices on a rectangle whose width is:

[MATH]w=\sqrt{2}r=10[/MATH]
And whose height is $h$ we've already found. Now we just need to use the Pythagorean theorem to write:

[MATH]\overline{AB}=\sqrt{w^2+h^2}=?[/MATH]

 

[Dr.Peterson]

You seem to have found the distance from A to B as measured on the curved surface of the cylinder. What it asks for is the straight-line distance between them (through the body of the figure).

 

[janemercado]

Oh I understand now, I was not finding the shortest distance between the two points. The only thing I am unclear about at this point, is how you found the width of the rectangle by multiplying the radius by the square root of 2? Is there a formula for this, or at least an explanation of why this calculation works?

 

[MarkFL]

Consider a right triangle whose legs are both $s$ in length (a right isosceles triangle). What is the length of the hypotenuse?

 

[janemercado]

Consider a right triangle whose legs are both $s$ in length (a right isosceles triangle). What is the length of the hypotenuse?

The length of the hypotenuse would simply be s*radical 2

 

[MarkFL]

The length of the hypotenuse would simply be s*radical 2

Yes, it was that result I used as the width of our rectangle is the hypotenuse of the right isosceles triangle having legs of length $r$.

 

[janemercado]

Ah okay, I understand it now after drawing it out. Now knowing that the width is 10 units and the height is 24 units, they can be put into the pythagorean theorem equation to come out with a c (hypotenuse) value of 26 units. Thank you very much!! It is very appreciated