The whole surface area of a regular pyramid is 340 sq.cm and the area of its base is 100 sq.cm. If the area of each congruent lateral face is 30 sq.cm, how many lateral faces does the pyramid have?

Introduction / Context:
This problem deals with the surface area of a regular pyramid. The pyramid has a regular polygon as its base and congruent isosceles triangles as its lateral faces. You know the total surface area, the base area, and the area of one lateral face. The goal is to find the number of such lateral faces, which is equal to the number of sides of the base polygon.

Given Data / Assumptions:
• Whole surface area of the pyramid is 340 sq.cm.
• Area of the base is 100 sq.cm.
• Area of each lateral triangular face is 30 sq.cm.
• All lateral faces are congruent, as the base is a regular polygon.

Concept / Approach:
Total surface area of a pyramid equals base area plus sum of areas of all lateral faces. If there are n lateral faces each of area 30 sq.cm, then total lateral area is 30n. Using the given total surface area and base area, we can solve for n directly with a simple linear equation.

Step-by-Step Solution:
Step 1: Write the relation for total surface area S. Step 2: S = base area + lateral area = 100 + 30n. Step 3: Given that S = 340 sq.cm, so 100 + 30n = 340. Step 4: Subtract 100 from both sides: 30n = 240. Step 5: Divide both sides by 30: n = 240 / 30 = 8. Step 6: Thus there are 8 congruent lateral faces, so the base is an octagon.

Verification / Alternative check:
If there are 8 faces, total lateral area is 8 × 30 = 240 sq.cm. Adding the base area, 240 + 100 = 340 sq.cm, which matches the given whole surface area. Therefore the count of lateral faces is validated.

Why Other Options Are Wrong:
• 7 faces give lateral area 210, total 310, which is less than 340.
• 9 faces give lateral area 270, total 370, which is greater than 340.
• 10 faces give lateral area 300, total 400, again not matching 340.

Common Pitfalls:
Some learners may confuse lateral area with whole surface area or inadvertently add the base area twice. Keeping a clear distinction between base area and lateral area and writing down the equation step by step avoids such errors.

Final Answer:
The pyramid has 8 lateral faces.