Question : The total surface area of a square-based right pyramid is 1536 m2, of which 37.5% is the area of the base of the pyramid. What is the volume (in m3) of this pyramid?
Option 1: 3048
Option 2: 3072
Option 3: 3144
Option 4: 3108
Correct Answer: 3072
Solution :
Given: Total surface area of pyramid = 1536 m
2
Area of base = 37.5% of total surface area
= $\frac{37.5}{100}\times1536$
= $\frac{3}{8}\times1536$
= $576$
Remaining area = 1536 – 576 = 960
$\therefore$ lateral surface area = 960 m
2
Area of square base = 576
⇒ $\text{side}^2=576$
$\therefore \text{side}=24$
Perimeter of base = 4 × 24 = 96
We know that,
Lateral surface area of pyramid = $\frac{1}{2}$ × perimeter of base × slant height
⇒ $960 = \frac{1}{2} \times 96\times$ slant height
$\therefore$ Slant height = 20
$\text{Height} = \sqrt{\text{slant height}^2-(\frac{\text{side}}{2})^2}$
= $\sqrt{20^2-12^2}$
= $16$
Now, Volume of pyramid = $\frac{1}{3}$ × area of base × height
= $\frac{1}{3}\times576\times16$
= $3072$ m
3
Hence, the correct answer is 3072.
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