Final answer:
To find the vertical height of the pyramid, subtract the base area from the total surface area to find the total triangular sides area, divide by four for one triangle's area, then use the triangle area formula to solve for the height. The height is found to be 17cm.
Step-by-step explanation:
The student is asking how to find the vertical height of a square-based pyramid whose total surface area is 1920cm² and whose base side is 30cm. To find the height, we first need to calculate the area of the base, which is the side length squared (since it's a square base). Then, we subtract the base area from the total surface area to find the total area of the four triangular faces. After that, we divide this by four to find the area of one triangular face and use the formula for the area of a triangle to solve for the pyramid's height.
First, calculate the area of the base:
Base area = side² = 30cm x 30cm = 900cm².
Next, subtract this from the total surface area to get the total area of the triangular sides:
Triangular sides area = Total surface area - Base area = 1920cm² - 900cm² = 1020cm².
Since there are four triangles, the area of one triangle is:
Area of one triangle = Triangular sides area ÷ 4 = 1020cm² ÷ 4 = 255cm².
Now, use the formula for the triangle area to solve for the height, where the base is the side of the square:
Triangular area formula = 1/2 × base x height.
Hence,
255cm² = 1/2 × 30cm x height.
Solving fo height gives us the vertical height of the pyramid:
Height = 2 × 255cm² / 30cm = 17cm.
Therefore, the vertical height of the pyramid is 17cm.