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Math Help - Least possible length - 2 cones in box

  1. #1
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    Least possible length - 2 cones in box

    Hi All,

    Almost(but not quite!) able to visualize this problem correctly.

    The ice cream is in the shape of a cone of base radius 4 cm and vertical height 12 cm. The curved surface is completely covered with wafer.

    (a) If one ice cream is packed with the base of the box. Calculate the least possible height of the box, correct to nearest mm.

    (b) If 2 ice creams are packed in a thin rectangular box with area of cross-section equal to 64 sq. cm, calculate the least possible length of the box, correct to the nearest mm.

    For (a), I calculated the angle at the vertex of the cone using tan. Then angle is twice that, = 36.9, and hence when viewed sideways slant height x sin 36 gives least height = 76 mm.

    This is the part I get.

    But I am stumped on the (b) section. Please advice.

    Thanks.
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  2. #2
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    Quote Originally Posted by mathguy80 View Post
    Hi All,

    Almost(but not quite!) able to visualize this problem correctly.

    ...

    (b) If 2 ice creams are packed in a thin rectangular box with area of cross-section equal to 64 sq. cm, calculate the least possible length of the box, correct to the nearest mm.

    ...
    I've attached a sketch how I would pack the cones - as far as I understand the question.
    Attached Thumbnails Attached Thumbnails Least possible length - 2 cones in box-coneinbox.png  
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  3. #3
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    That's what I thought, mirrored cones, but what significance does the cross section have then? I am confused with the phrasing of this problem too.
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  4. #4
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    Quote Originally Posted by mathguy80 View Post
    That's what I thought, mirrored cones, but what significance does the cross section have then? I am confused with the phrasing of this problem too.
    A circle with r = 4 cm - that means a diameter of 8 cm - fits exactly into a square with side length 8 cm. The area of such a square is A = 64 cm².

    If you take a box with a base area of 8 cm by 8 cm and you tilt the first cone a little bit (as shown in my sketch) then the 2nd cone will slide down a little bit (how much?) so that the height of the box could be reduced a little bit. But that's a lot of little bits and a lot calculating for only 2 ice cones .
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  5. #5
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    Ha!, I see it now. Yup a lot of work for 2 ice cones.
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