Problem #29

Posted on April 1, 2010 | 1 Comment

A regular hexagonal pyramid has one of its lateral edges four times the length of one side of the base. If the volume of the pyramid is 60 cubic centimeters, compute the number of centimeters in the length of one side of the base in simplest radical form.

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One Response to Problem #29

Sterling | April 9, 2010 at 20:23 | Reply

Let the side length of the base be $s$ . Then the area of the base is $6[s^2 \sqrt{3} /4] = 3s^2 \sqrt{3} /2$ .

If the lateral edge has length $4s$ , then the height of the pyramid is given by the Pythagorean Theorem: $h = \sqrt{(4s)^2 - s^2} = s\sqrt{15}$ .

The volume of the pyramid is $B\cdot h/3$ , where B is the area of the base.

Hence $60 = s^3\cdot 3\sqrt{5} / 2$ , so $s^3 = 40/\sqrt{5} = 8\sqrt{5}$ . Thus the side length is $2\sqrt[6]{5}$ .