# 010 Review Problem - Volume of a stick

**Problem 10**

How many cubic inches of lumber does a stick contain if it is 4 in. by 4 in. at one end, 2 in. by 2 in. at the other end, and 16 ft. long?

**Solution 10**

Use the formula for volume of prismatoid

$V = \frac{1}{6}L(A_1 + 4A_m + A_2)$

$A_1 = 4 \times 4 = 16 ~ \text{in.}^2$

$V = \frac{1}{6}L(A_1 + 4A_m + A_2)$

$A_2 = 2 \times 2 = 4 ~ \text{in.}^2$

$A_m = \dfrac{4 + 2}{2} \times \dfrac{4 + 2}{2} = 9 ~ \text{in.}^2$

$L = 16(12) = 192 ~ \text{in.}$

Thus,

$V = \frac{1}{6}(192)[ \, 16 + 4(9) + 4 \, ]$

$V = 1792 ~ \text{in.}^3$ *answer*

**Another Solution**

Use the formula for frustum

$V = \frac{1}{3}(A_1 + A_2 + \sqrt{A_1 \, A_2}) \, h$$A_1 = 4 \times 4 = 16 ~ \text{in.}^2$

$V = \frac{1}{3}(A_1 + A_2 + \sqrt{A_1 \, A_2}) \, h$

$V = \frac{1}{3}(A_1 + A_2 + \sqrt{A_1 \, A_2}) \, h$

$A_2 = 2 \times 2 = 4 ~ \text{in.}^2$

$h = 16(12) = 192 ~ \text{in.}$

Thus,

$V = \frac{1}{3}[ \, 16 + 4 + \sqrt{16(4)} \, ](192)$

$V = 1792 ~ \text{in.}^3$ *answer*

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