Question

1. a gem is cut in the shape of two square pyramids that are joined at their base. the length of each side of the base is 10cm and the height of each pyramid is 12cm. determine the surface area of the composite solid.

a) 520 cm²
b) 544 cm²
c) 619 cm²
d) 729 cm²

Explanation:

Step1: Find slant height of pyramid

The base side is \( 10 \, \text{cm} \), so half - base side \( r=\frac{10}{2} = 5 \, \text{cm} \), height \( h = 12 \, \text{cm} \). Using Pythagorean theorem for slant height \( l \): \( l=\sqrt{r^{2}+h^{2}}=\sqrt{5^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13 \, \text{cm} \).

Step2: Find lateral surface area of one pyramid

Lateral surface area of a square pyramid is \( L=\frac{1}{2}Pl \), where \( P \) is perimeter of base. \( P = 4\times10=40 \, \text{cm} \), \( l = 13 \, \text{cm} \). So \( L=\frac{1}{2}\times40\times13 = 260 \, \text{cm}^2 \).

Step3: Find total surface area of composite solid

The composite solid has two square pyramids joined at their bases, so we don't include the area of the base (since it's internal). Total surface area \( S=2\times260=520 \, \text{cm}^2 \)? Wait, no, wait. Wait, the formula for lateral surface area of one pyramid is \( 4\times(\frac{1}{2}\times10\times13)=4\times65 = 260 \). Two pyramids: \( 2\times260 = 520 \)? But wait, let's re - check. Wait, the slant height calculation: \( r = 5 \), \( h = 12 \), \( l=\sqrt{5^{2}+12^{2}} = 13 \). Then area of one triangular face is \( \frac{1}{2}\times10\times13 = 65 \). There are 4 triangular faces per pyramid, so one pyramid's lateral surface area is \( 4\times65=260 \). Two pyramids: \( 2\times260 = 520 \). But wait, the options have 520 as option A. But wait, maybe I made a mistake? Wait, no, the composite solid is two square pyramids joined at their bases, so the total surface area is the sum of the lateral surface areas of both pyramids (since the base areas are glued together and not part of the surface area). So each pyramid has 4 triangular faces, each with area \( \frac{1}{2}\times10\times13 = 65 \). So one pyramid: \( 4\times65 = 260 \), two pyramids: \( 2\times260=520 \).

Answer:

A) \( 520 \, \text{cm}^2 \)