a plastic container company design a container with a lid to hold one piece of pie

Question

a) design the container as a traingular prism.explain your choice of dimension b) calculate the area of plastic in your design ( in the figure the height of container is 5 cm and length is 23 cm and 6 pieces of pie

Answer 1

Let the radius of the circular pie be r and the thickness t. Also let the number of slices be N. We can then find a formula for the dimensions of the container. The length of the isosceles triangle forming the base and lid of the container is the height of the triangle, which is equal to r. The angle of the isosceles triangle is 360/N.

To find the length of the isosceles triangle's base, drop a perpendicular from the apex of the triangle on to its base. The perpendicular will bisect the apex angle, so we have two back to back right-angled triangles with one angle equal to 360/2N=180/N. The half-length of the base of the triangle=rtan(180/N). So the base of the triangle is 2rtan(180/N). We then need to decide how many slices we are going to put into the container. Let S be the number of slices. So the height of the container is St at least to accommodate the slices.

Now we can calculate the area of the prism. It has 3 rectangular sides and 2 triangles for top and bottom. The long side of the container has length L given by r/L=cos(180/N) so L=r/cos(180/N). 

Rectangular areas: StL=rSt/cos(180/N) (two of these), base * thickness=2rtan(180/N)*St=2rSttan(180/N).

Triangular base and lid: r^2tan(180/N) (two of these).

The total area is the sum of these 5 shapes: 2r(St(sec(180/N)+tan(180/N))+2rtan(180/N)).

If the container is to hold the entire pie then S=N.

Now put in some figures: r=23cm, t=0.8cm, N=S=6. 180/N=30. tan30=0.5774, cos30=0.8660. St=4.8cm.

More to follow...

Comments

Its too complicated . Its a grade 8 question

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I had no idea, sorry, and there was no picture given with the question. I'll leave it as it is because I don't know what else I can do. If the dimensions to work with are presented, I think I could produce a much simpler solution. I just gave a general solution. Is the triangular section equilateral?

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Actually thereisnt any option to place a pix . I uploaded the question pix on yahoo ans too but havent got ans yet :( anywaz u did work hard thanx 4 it . If its possible 4 u to give me ur id so i can mail u question please

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I'd be happy to help. There is a facility for uploading photos into a question, solution or comment, if you can use that. But there was only one fact missing: the third side of the triangular base and top. If you go to my profile page by clicking on my name you can send me a private message. If you include your email address, we can communicate that way. I'm not prepared to disclose my email address in an open comment for all to see! If grade 8 includes finding the area of a triangle knowing its base and height, there is indeed a simpler solution. All we need is the base length.

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The picture shows the circular pie with a diameter of 23cm, radius 11.5cm. The 60º sector represents the piece of pie that has to fit in the container. If you draw this to scale you'll find you need to extend the sides of the sector another 2cm approximately so that the piece of pie will fit neatly into a prism 5cm high (the thickness of the pie) so that the edge of the pie isn't cut off. The thin red lines show where the triangle forming the lid and base of the container has to be extended from the sector arms. So you have a 13.5cm equilateral triangle as the top and bottom of the prism. The dotted line shows where the pie would be cut off if you only made the triangle side 11.5cm, the radius of the pie.

Answer 2

23cm