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 = 1/3h(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 ft^3. (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: Identify the formula and values

We are given the volume - formula for the frustum of a square pyramid $V=\frac{1}{3}h(a^{2}+ab + b^{2})$, where $h = 206$ ft, $b = 760$ ft, and $a = 318$ ft.

Step2: Calculate $a^{2}$, $ab$, and $b^{2}$

$a^{2}=318^{2}=318\times318 = 101124$
$ab=318\times760 = 241680$
$b^{2}=760^{2}=760\times760 = 577600$

Step3: Calculate $a^{2}+ab + b^{2}$

$a^{2}+ab + b^{2}=101124+241680 + 577600=920404$

Step4: Calculate the volume $V$

$V=\frac{1}{3}\times206\times920404$
$V=\frac{206\times920404}{3}$
$206\times920404 = 189603224$
$V=\frac{189603224}{3}\approx63201075$ ft³

Answer:

$63201075$