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Repeating Question, Need Help (Sorry Guys)

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A metal washer has an external diameter of 2cm and an internal diameter of 1cm.

a. How many washers can be cut from a sheet of steel 3m by 1m?

b. If the metal left over was remelted and cast into a sheet of the same thickness and width 1m, how many additional washers could be cut?

The answer to "a" is 7500 washers.

I am having a problem with "b." The answer to that is 3050 but I do not know the process and can't fully understand the question. Please provide a detailed explanation of the process and question. (The answer has to be 3050)

Note the units. They can be tricky.

Thanks! :)

Jun 26, 2018

#1
+106533
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Here is my attempt, Ghosty.. however..I don't come up with  "3050"

Assuming  a uniform thickness of the sheets  throughout...

We have 7500 washers .......

Note that the amount of leftover metal that is cut from the center of each washer  is just

pi (inside radius)^2   =  pi (1/2 cm)^2  =  pi/ 4  cm^2

So.....the amount of this excess metal for 7500 washers is just  7500 (pi/ 4)cm^2  =

1875 pi cm^2    (1)

However....we must also figure the additional amouint of excess metal on the sheet that surrounds all of the washers

Note that the area of the sheet is just  (300cm  * 100cm)  = 30000cm^2

And the are occupied by each washer is just  pi (outside radius)^2  = pi (1 cm)^2  = pi  cm^2

So......7500 washers occupy  7500pi  cm^2

So....the excess metal  surrounding all the washers is  just

Area of the sheet  - area  occupied by the washers =

(30000  - 7500pi) cm^2   (2)

So....the total amount of excess metal available  for forming new washers  is just  the sum of   (1)  and (2)  =  [  1875 pi  +  (30000 - 7500pi) ] =  [30000 - 5625 pi ]  cm^2

So....the total number of   possible washers   that can be made with a new sheet   is just

[  Area  of metal available  ]  / [ Area  occupied by each washer ]   =

[ 30000 - 5625pi ] cm^2  / [ pi] cm^2  ≈  3924 additional washers

Jun 26, 2018
#2
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Thanks CPhill

So sorry, but could you please explain the final part of your explanation again? I'm trying to fully grasp the concept.

This is (part of) what you said:

"So....the excess metal  surrounding all the washers is  just

Area of the sheet  - area  occupied by the washers =

(30000  - 7500pi) cm^2   (2)

So....the total amount of excess metal available  for forming new washers  is just  the sum of   (1)  and (2)  =  [  1875 pi  +  (30000 - 7500pi) ] =  [30000 - 5625 pi ]  cm^2

So....the total number of   possible washers   that can be made with a new sheet   is just

[  Area  of metal available  ]  / [ Area  occupied by each washer ]   =

[ 30000 - 5625pi ] cm^2  / [ pi] cm^2  ≈  3924 additional washers"

I just need help on the bolded part. Especially [pi] cm^2.

Thanks again.

Gh0sty15  Jun 26, 2018
edited by Gh0sty15  Jun 26, 2018
#3
+106533
+3

OK

When we cut all the washers out of the sheet   (  and before we cut any holes in the washers )  the area occupied by each washer  is just  pi  ( outside radius)^2  = pi (1 cm)^2  = pi cm^2

[Note that if the diameter of each washer is 2 cm....then the radius of each is just 1 cm  ]

So....the area occupied by each washer is that of a circle with a radius of 1 cm  =

pi (radius)^2  = pi  (1 cm)^2   =  pi  cm^2

So.....the total are occupied by these washers is  just  7500* pi  cm^2

And the area of the sheet  is  just  (300 cm * 100 cm)  = 30000 cm^2

So....the amount of excess metal surrounding the washers   is just the total area  of the sheet  less the total area occupied by all the washers   = [ 30000  - 7500 pi ] cm^2    (1)

When we take all the washers off the original sheet  [ 7500 of them ] we will cut the centers out  of each of them.....the amount of metal in the center  (the area of metal that will form the hole in the washer )  is just  pi * ( inside radius)^2  = pi  (1/2 cm)^2  = pi/4 cm^2

Since we have 7500 of these....the total area of excess metal from this part is just

7500 *[ pi /4 cm^2]  =   1875 pi  cm^2     (2)

So  the total amout of metal  that we have  to cut washers from is now just  the sum of (1)  and (2)  =

[30000 cm^2  - 7500 pi cm^2]  + 1875 pi cm^2  =  30000cm^2  - 5625 pi cm ^2  =

[ 30000  - 5625 pi ] cm^2   = area of the new sheet

And each washer  cut from this sheet will  occupy the same area as before  = pi cm^2

So....the total number   of washers we can cut from the new sheet  is

[ Area  of new sheet  ] / [ Area of each washer ] =  [ 30000 - 5625pi] cm^2  /  pi  cm^2   =

[ 30000  - 5625 pi ]  / pi   =

Does that make sense  ??

CPhill  Jun 26, 2018
#4
+581
+2

Yeah, I get it now.

Thanks, a lot CPhil.

The answer sheet I have to solve this question had many flaws, so I guess it was wrong again (it often is...)

Your explanation was great! Really thorough and detailed.

Thanks again.

Gh0sty15  Jun 26, 2018