**Problem 27**

One hundred and fifty posts are used in fencing a lawn. Each post is built in the form of frustum of a pyramid surmounted by a pyramid whose lower base is common with the upper base of the frustum. The height of the pyramidal top is 2 in. and the common base is a square 4 in. on an edge. The lower base of the frustum has an edge of 6 in. If the overall height of each is 6 ft., how much concrete will be used in making the posts?

**Solution 27**

$V_p = \frac{1}{3}A_bh = \frac{1}{3}(\frac{4}{12})^2(\frac{2}{12})$

$V_p = \frac{1}{162} ~ \text{ft.}^3$

Volume of frustum part

$V_f = \frac{1}{3}(A_1 + A_2 + \sqrt{A_1A_2})h$

$V_f = \frac{1}{3}[ \, (\frac{6}{12})^2 + (\frac{4}{12})^2 + \sqrt{(\frac{6}{12})^2(\frac{4}{12})^2} \, ] (6 - \frac{2}{12})$

$V_f = \frac{665}{648} ~ \text{ft.}^3$

Volume of 1 post

$V_1 = V_p + V_f = \frac{1}{162} + \frac{665}{648}$

$V_1 = \frac{223}{216} ~ \text{ft.}^3$

Volume of 150 posts

$V = 150V_1 = 150(\frac{223}{216})$

$V = 154.86 ~ \text{ft.}^3$ *answer*

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