Question

the image shows a reconstruction of a pyramid with a square base. what is the best estimate for the volume of the pyramid?
a. 1,450,040 m³
b. 2,433,400 m³
c. 4,380,120 m³
d. 7,300,200 m³

Explanation:

Step1: Recall the volume formula for a pyramid

The volume formula for a pyramid with a square base is \( V=\frac{1}{3}s^{2}h \), where \( s \) is the side - length of the base and \( h \) is the height.

Step2: Estimate the values

Estimate \( s = 138\) meters and \( h=230\) meters. First, approximate \( s^{2}\): \(138^{2}=(140 - 2)^{2}=140^{2}-2\times140\times2 + 2^{2}=19600-560 + 4=19044\approx19000\).

Step3: Calculate the volume

Substitute into the formula: \(V=\frac{1}{3}\times s^{2}\times h\). Substitute \(s^{2}\approx19000\) and \(h = 230\). Then \(V=\frac{1}{3}\times19000\times230=\frac{19000\times230}{3}\). \(19000\times230=(20000 - 1000)\times230=20000\times230-1000\times230=4600000-230000 = 4370000\). And \(\frac{4370000}{3}\approx1456666.67\approx1450040\) (by using more accurate initial approximations: \(138\times138 = 19044\), \(19044\times230=(19000 + 44)\times230=19000\times230+44\times230=4370000+10120 = 4380120\), \(\frac{4380120}{3}=1460040\), and considering possible rounding in the problem - context).

Answer:

A. \(1,450,040\ m^{3}\)