To find the volume of the prism, we need to follow these steps:
- Determine the missing leg of the triangle.
- Calculate the area of the triangular base.
- Find the volume of the prism using the base area and the given height of the prism.
Step 1: Determine the Missing Leg
Given:
[tex]\large\bullet\ \ \text{One leg of the triangle, $a=$18 inches}[/tex]
[tex]\large\bullet\ \ \text{Hypotenuse, $c=$30 inches}[/tex]
[tex]\large\text{We need to find the other leg, $b.$}[/tex]
[tex]\large\text{Using the Pythagorean theorem:}[/tex]
[tex]\large\text{$a^2+b^2=c^2$}[/tex]
[tex]\large\text{Substitute the known values:$}[/tex]
[tex]\large\text{$18^2+b^2=30^2$}\\[/tex]
[tex]\large\text{$324+b^2=900$}[/tex]
[tex]\large\text{$b^2=576$}[/tex]
[tex]\large\text{$b=\sqrt{576}$}[/tex]
[tex]\large\text{$b=24$}[/tex]
So the other leg of the triangle is 24 inches.
Step 2: Calculate the Area of the triangular base
[tex]\large\text{The area $A$ of a right triangle is given by:}[/tex]
[tex]\large\text{$A=\dfrac{1}{2}\times$base$\times$height}[/tex]
[tex]\large\text{Here, $a$ and $b$ are the height and base of the triangle:}[/tex]
[tex]\large\text{$A=\dfrac{1}{2}\times18\times24$}[/tex]
[tex]\large\text{$A=216$ square inches}[/tex]
Step 3: Find the volume of the prism
[tex]\large\text{The volume $V$ of a prism is given by:}\\[/tex]
[tex]\large\text{$V=$Base Area $(A)\times$ height (length) of the prism}[/tex]
[tex]\large\text{Calculate the volume:$}[/tex]
[tex]\large\text{$V=216\times15=3240$ cubic inches}[/tex]
