Question

1. find the volume of the rectangular prism. (10 points) (volume = l·w·h) volume 4. find the area of the shaded region. (12 points)

Explanation:

Step1: Recall volume formula for rectangular prism

The volume formula of a rectangular prism is $V = l\times w\times h$. Here, $l = 2x + 3$, $w=3x - 1$, and $h = 5x$.

Step2: Multiply the expressions

First, multiply $(2x + 3)$ and $(3x - 1)$ using the FOIL method:
$(2x+3)(3x - 1)=2x\times3x-2x\times1 + 3\times3x-3\times1=6x^{2}-2x + 9x-3=6x^{2}+7x - 3$.
Then multiply the result by $5x$:
$V=(6x^{2}+7x - 3)\times5x=6x^{2}\times5x+7x\times5x-3\times5x = 30x^{3}+35x^{2}-15x$.

Step3: Recall area - subtraction formula for the shaded region

The area of the large rectangle is $A_{1}=3x(3x + 5)=9x^{2}+15x$.
The area of the small rectangle is $A_{2}=x(x + 3)=x^{2}+3x$.

Step4: Calculate the area of the shaded region

Subtract the area of the small rectangle from the area of the large rectangle:
$A=(9x^{2}+15x)-(x^{2}+3x)=9x^{2}+15x - x^{2}-3x=8x^{2}+12x$.

Answer:

Volume: $30x^{3}+35x^{2}-15x$
Area of shaded region: $8x^{2}+12x$