Question

1. find an expression for the volume of a triangular prism where the base of the prism is a right triangle with base (4x + 6) and height (x + 5) and the height of the triangular prism is (x + 5). use the expression to find the volume when (x = 3).

Explanation:

Step1: Recall the volume formula for a triangular prism

The volume \( V \) of a triangular prism is given by \( V=\frac{1}{2}\times\text{base of triangle}\times\text{height of triangle}\times\text{height of prism} \). Here, the base of the triangle is \( 4x + 6 \), the height of the triangle is \( x + 5 \), and the height of the prism is \( x + 5 \). So the volume formula becomes:
\( V=\frac{1}{2}(4x + 6)(x + 5)(x + 5) \)
Simplify \( \frac{1}{2}(4x + 6) \) first. Factor out 2 from \( 4x+6 \): \( 4x + 6=2(2x + 3) \), so \( \frac{1}{2}\times2(2x + 3)=(2x + 3) \). Then the volume formula is \( V=(2x + 3)(x + 5)^2 \)

Step2: Expand \( (x + 5)^2 \)

Using the formula \( (a + b)^2=a^2+2ab + b^2 \), where \( a = x \) and \( b = 5 \), we get \( (x + 5)^2=x^2+10x + 25 \)

Step3: Multiply \( (2x + 3) \) with \( (x^2+10x + 25) \)

Using the distributive property (FOIL for polynomials):
\( (2x+3)(x^2 + 10x+25)=2x(x^2 + 10x+25)+3(x^2 + 10x+25) \)
\( =2x^3+20x^2 + 50x+3x^2+30x + 75 \)
Combine like terms: \( 2x^3+(20x^2+3x^2)+(50x + 30x)+75=2x^3+23x^2+80x + 75 \)

Step4: Substitute \( x = 3 \) into the volume formula

First, find the base of the triangle when \( x = 3 \): \( 4x+6=4\times3 + 6=12 + 6 = 18 \)
Height of the triangle: \( x + 5=3 + 5 = 8 \)
Height of the prism: \( x + 5=3 + 5 = 8 \)
Now use the volume formula \( V=\frac{1}{2}\times\text{base}\times\text{height of triangle}\times\text{height of prism} \)
\( V=\frac{1}{2}\times18\times8\times8 \)

Step5: Calculate the volume

First, \( \frac{1}{2}\times18 = 9 \)
Then, \( 9\times8\times8=9\times64 = 576 \)

Answer:

The volume of the triangular prism when \( x = 3 \) is \( 576 \)