Really hard modelling and problem solving calculus question!!!

[LBrenda16]

A light-weight "pop-up" tent consists of six flexible plastic struts that are inserted into pockets sewn into the joins of the fabric panels. The resulting shape has hexagonal horizontal cross-sections, while vertical cross-sections through the centre are semi-circular.

Derive a formula for the volume of the tent as a function of its height.

How do i do it? I tried doing a symmetrical trapezium on the graph and revolved it around half way to get the tent look, not a sphere, but the top part (the roof of the tent) was in the shape of a trapezium (shaped like a diamond? like theres edges), not half a sphere like it should be, and the base being a hexagon. If you can imagine it? Anyways, need help!

Attached is the question and a picture.

 

[Deleted member 4993]

A light-weight "pop-up" tent consists of six flexible plastic struts that are inserted into pockets sewn into the joins of the fabric panels. The resulting shape has hexagonal horizontal cross-sections, while vertical cross-sections through the centre are semi-circular.

Derive a formula for the volume of the tent as a function of its height.

How do i do it? I tried doing a symmetrical trapezium on the graph and revolved it around half way to get the tent look, not a sphere, but the top part (the roof of the tent) was in the shape of a trapezium (shaped like a diamond? like theres edges), not half a sphere like it should be, and the base being a hexagon. If you can imagine it? Anyways, need help!

Attached is the question and a picture.

Use "differential" slices parallel to the base (which is hexagonal - I assume regular hexagon i.e. each sides and included angles are congruent).

The side of the hexagon would be equal to the radius of the circumscribing circle (which can be found from the description of the hemispherical dome).

Now find the limits (again from the description of the hemispherical dome) and integrate.

 

[HallsofIvy]

You are told that "The resulting shape has hexagonal horizontal cross-sections" so an obvious thing to do would be to take thin "slabs" as the area of the hexagon times dh. As for the size of the hexagons, you are told that "vertical cross-sections through the centre are semi-circular" meaning that the struts form a sem-circle and the height of the tent is a radius while the distance between opposite vertices of the base is a diameter. You are asked to find the volume of the tent as a function of its height, so call that "H". At the base, the area is a hexagon with "diameter" (distance between opposite vertices) 2H. What is the area of that? At a given height h (0 to H), That "diameter" is the distance across the semicircle at height h. What is that?