Question

an amazing formula from ancient mathematics was used by the egyptians to find the volume of the frustum of a square pyramid, as shown in the figure. its volume is given by v = \frac{1}{3}h(a^{2}+ab + b^{2}), where b is the length of the base, a is the length of the top, and h is the height.
a. when a pyramid was partially completed to a height h of 206 ft, b was 760 ft, and a was 318 ft. calculate its volume at this stage of construction.
the volume is 63201075 ft³. (round to the nearest integer as needed.)
b. try to visualize the figure if a = b. what is the resulting shape? find its volume.
the resulting shape is rectangular box with a square base.
the volume of the obtained figure is v = . (simplify your answer.)

Explanation:

Step1: Substitute a = b into volume formula

Given $V=\frac{1}{3}h(a^{2}+ab + b^{2})$, when $a = b$, we have $V=\frac{1}{3}h(a^{2}+a\times a+a^{2})$.

Step2: Simplify the expression

$V=\frac{1}{3}h(3a^{2})$.

Step3: Further simplify

$V = a^{2}h$.

Answer:

$a^{2}h$