Question

determining missing lengths
in the right triangular prism below, \\(\overline{bg} \perp \overline{fc}\\)
the volume of the prism is 900 cubic centimeters. the height of the prism is 15 centimeters and the height of the triangular base is 10 centimeters. what is the length of edge fc?
\\(\square\\) cm

Explanation:

Step1: Recall prism volume formula

The volume \( V \) of a prism is \( V = B \times h_p \), where \( B \) is the area of the triangular base, and \( h_p \) is the height of the prism.

Step2: Solve for base area \( B \)

Rearrange the formula to \( B = \frac{V}{h_p} \). Substitute \( V = 900 \, \text{cm}^3 \), \( h_p = 15 \, \text{cm} \):
\( B = \frac{900}{15} = 60 \, \text{cm}^2 \)

Step3: Recall triangle area formula

The area of a triangle is \( B = \frac{1}{2} \times b \times h_t \), where \( b = FC \) (the base of the triangle), and \( h_t = 10 \, \text{cm} \) (height of the triangular base).

Step4: Solve for \( FC \)

Rearrange to \( b = \frac{2B}{h_t} \). Substitute \( B = 60 \, \text{cm}^2 \), \( h_t = 10 \, \text{cm} \):
\( FC = \frac{2 \times 60}{10} = 12 \, \text{cm} \)

Answer:

12