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! :)

Gh0sty15
Jun 26, 2018

#1**+3 **

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

Your answer to "a" is correct !!!

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

CPhill
Jun 26, 2018

#2**+2 **

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

#3**+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 =

about 3924 washers

Does that make sense ??

CPhill
Jun 26, 2018