Let Vp be the volume of the square-based pyramid with the same base as the tent and has its top vertex at the apex of the tent. Let Vb be the volume of the square-based box surrounding the tent, and let V (as above) be the volume of the tent, all volumes measured in cubic feet. Compute the double ratio Vp : V : Vb. Does it seem reasonable?
The base of a garden tent is a square with side
length c feet. Two semi-circular poles run between pairs of opposite vertices of
the base and support the tent. Cross-sections parallel to the base are squares.
Compute the volume enclosed by the tent as follows:
(a) Establish a rectangular
the tent’s base, the positive y-axis pointing directly upward through the tent’s
apex, and the postive x-axis passing through a vertex of the base. Sketch the
tent with this coordinate system. Label the axes with their variable names
and units.
(b) Sketch the cross-section of the tent cut out by the xy-plane (a semi-circle)
in the xy-coordinate system. Label the coordinates of the intercepts of the
semi-circle. Label the axes with their variable names and units.
(c) Sketch a general horizontal segment with its endpoints P and Q on the
semicircle in part (b). Label the coordinates of its endpoints P and Q as
(−x, y) and (x, y), respectively. Assume x > 0.
(d) Sketch the square cross-section the tent parallel to the base that passes
through P and Q. Label the points P and Q in your sketch.
(e) Express the area A of the cross-section in part (d) first in terms of x and
then in terms of y. Include units.
f ) Express the differential volume dV of a slice y feet above the base in terms of y and the differential thickness of the slice. Include units. Sketch the slice.
(g) Set up a definite
(h) Evaluate the definite integral to find an expression for the volume enclosed by the tent in terms of c.
(i) Check that your expression for the volume has the correct units.
(j) Let Vp be the volume of the square-based pyramid with the same base as the tent and has its top vertex at the apex of the tent. Let Vb be the volume of the square-based box surrounding the tent, and let V (as above) be the volume of the tent, all volumes measured in cubic feet. Compute the double ratio Vp : V : Vb. Does it seem reasonable?
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