Question

B) If the volume of the prism given aside is 432 cm³, find its height.​

Answer 1

18 cm

Explanation:

The given diagram shows a triangular prism with right triangular bases ABC and A'B'C'. The height of the prism is the perpendicular distance between the two parallel and congruent bases. This height can be measured along lines AA', BB', or CC'.

The volume of a triangular prism can be calculated by multiplying the area of one of the triangular bases by the height of the prism. Therefore, to find the height of the prism, we can divide the volume by the area of one of the triangular bases.

The base of the given prism is a right triangle, with leg AB measuring 8 cm and the hypotenuse AC measuring 10 cm.

The area of a right triangle is equal to half the product of the lengths of its two legs. Therefore, to calculate the area of the triangular base (ABC), we must first find the length of leg BC. To do this, use the Pythagorean Theorem:

`\begin{aligned}AB^2+BC^2&=AC^2\\8^2+BC^2&=10^2\\64+BC^2&=100\\64+BC^2-64&=100-64\\BC^2&=36\\√(BC^2)&=√(36)\\BC&=6\; \sf cm\end{aligned}`

Now, calculate the area of triangle ABC by taking half of the product of its two legs, AB and BC:

`\begin{aligned}\textsf{Area of $\triangle ABC$} &= (1)/(2) \cdot AB \cdot BC\\\\&= (1)/(2) \cdot 8\; \textsf{cm}\cdot 6\; \textsf{cm}\\\\&= 4\; \textsf{cm}\cdot 6\; \textsf{cm}\\\\&= 24 \; \sf cm^2\end{aligned}`

To find the height of the prism, divide the given volume (432 cm³) by the area of triangular base ABC:

`\begin{aligned}\textsf{Height}&= \frac{\sf Volume}{\textsf{Area of $\triangle ABC$}}\\\\&= (432\; \sf cm^3)/(24\; \sf cm^2)\\\\&= 18 \sf \; cm\end{aligned}`

Therefore, the height of the prism is 18 cm.