# Two parallel shelves, each exactly 138 cm long, have exactly 32 cm of space between them...?

Two parallel shelves, each exactly 138 cm long, have exactly 32 cm of space between them. A circular hoop is placed between the shelves. The hoop's cross-section is a 2.5 cm square. The hoop barely touches the top shelf, and rests on the corners of the bottom shelf.

If the hoop's density is 7.85 grams per cubic centimetre, what is the hoop's total mass?

Relevance

Very interesting question. Get a cup of coffee or a coke: you're going to be with it a bit.

Step One

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You need to draw a diagram of this. Please follow my directions

Draw a Shelf. Label the left side A and the right side B. Label what you think is the midpoint as C

Draw the top shelf above the shelf you just drew. Call the left side D and the right side E. Call the Midpoint of this second shelf M.

Now draw a line through CM and extend it down the page. At some point stop and label the stopping point. O. O is the center of your circle.

Draw a circle that goes through AB and M.

Step Two

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Key step. There's many ways you could proceed but it is quickest to take the shortest way. Calculate the radius of the circle.

It turns out that for all circles the following is true.

AC^2 = CM*(2R - CM)

I'll see if I can translate that into language for you. Any chord of a circle (in this case AB) cuts a diameter (in this case OM*2) in a ratio such at the chord's midpoint the the diameter is cut into two pieces. These two pieces are the perpendicular distance from the circles circumference to the midpoint of the chord. The other piece of the diameter is from the midpoint of the chord (AB) to the opposite end of the circle through the center.

I know all of that sounds horrible. But just follow it with your diagram.

OK. Now we're ready to state or givens and find R.

R = ??

CM = 32 cm

AC = 1/2 of AB = 1/2 * 138 = 69 cm

Formula

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CM*(2R - CM) = AC^2

Solve

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32*(2R - 32) = 69^2 Substitute and remove brackets

64R - 1024 = 4761 Add 1024 to both sides

64R = 5785 Divide by 64

R = 5785/64

R = 90.39 cm

Step Three

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Find the circumference of the circle.

Formula

C = 2*pi*R

C = 2*3.14*90.39

C = 567.9 cm.

Step Four

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Take the hoop down from the shelves and make 1 clean cut in it with the sharpest knife you have. Stretch it out straight. It is a great big long cylinder now. Find the volume of it.

Volume = Cross sectional Area * the Circumference which is the length of the cylinder.

Volume = 2.5 cm^2 * 567.9 cm

Volume = 1419.9 cm^3

Step Five

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Find the mass.

d = m/V

V = 1414.9

d = 7.85 g/cm^3

m = ???

m = d * V

m = 7.85 * 1419.9

m = 11202.6

The mass of the ring is 11202.6 grams or 11.2 kg

Comment

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You don't have to follow all the gobbledygook that talks about the finding of R. Just try and follow the actual formula and the solution for R.