SOLUTION: A component (circle-shaped with a diameter measuring 60 cm) is stamped out of sheet steel. A square(with side measuring 30 cm) in the center of that circle is discarded. These comp

Click here to see ALL problems on Surface-area

Question 1161726: A component (circle-shaped with a diameter measuring 60 cm) is stamped out of sheet steel. A square(with side measuring 30 cm) in the center of that circle is discarded. These components are stamped out of a continuous steel coil with a width of 70 cm. The stamping process requires a gap of 25mm between each component. The steel coil is supplied in lengths of 25 meters costing $200.
a) What is the approximate percentage of steel wasted including the center square?
b) Assuming minimal wastage, how many components can be produced from each 25-meter coil?
c) What is the approximate cost of a component if the scrap is sold at 50% of the cost?

Found 2 solutions by solver91311, KMST:
Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

.

See diagram. There are exactly 40 pieces of the 25-meter coil of steel that measure 62.5 cm -- the diameter of the required circle plus the required 25 mm (2.5 cm) gap.

For one of the pieces, the scrap is the portion outside of the circle and inside of the green square. The total area of one piece is 62.5 x 70 square cm. The area of the circle is 30²π cm², and the area of the square is 30² cm². So the area of scrap is (62.5 x 70) - 30²π + 30² cm², which, divided by the total area of the piece gives the scrap percentage.

The scrap percentage calculated above times the $200 cost of the coil gives the cost of the scrap. Half of that cost subtracted from the $200 gives the total cost of material for the 40 components produced. So (200 - (Scrap Cost/2))/40 gives the per component material cost.

You can do your own arithmetic.

John

My calculator said it, I believe it, that settles it

Answer by KMST(5328) (Show Source):
You can put this solution on YOUR website!
b) You can stamp up to circles,
Stamping the 60cm diameter circles centered on the mid-width of the sheet, 25mm (2.5cm) apart from each other would allow us to stamp along the 25m (2500cm) steel coil
circles.
In fact circles with gaps between them, adding per gaps adds to a length of

and we could use the leftover length
to leave some space between the circles at the ends and the beginning and end of the coil.
Offsetting the circles so they are shifted a bit to the left and the right, makes the string of circles a little shorter, but not short enough to fit an additional circle.
The most you could offset centers is ,
and making the distance between the centers centers , would make the circles look like this
with and .
Each pattern with 2 circles (red rectangle) takes up
, so circles require out of the length of the coil.

a) The area of each circle is
The area of the square cut off is .
The area of each component stamped is

The total area of the stamped components is

The area of a sheet coil wide and long is
.
Out of that surface area turns into components, and the rest is "steel wasted including the center square".
The surface area of the steel wasted including the center square" is

As a percentage of the in the coil, that is

c) The of the coil wasted represents a cost of .
Selling it at of the cost reclaims ,
making the cost of components , and the cost per component