Question

example 5: find an expression for the volume of the shape below. simplify the expression, then find the actual volume if x = 3cm. image of a rectangular prism with length 2x², width 3x, height x³

Explanation:

Step1: Recall volume formula for rectangular prism

The volume \( V \) of a rectangular prism (cuboid) is given by the product of its length, width, and height. So, \( V=\text{length} \times \text{width} \times \text{height} \). Here, length \( = 2x^{2} \), width \( = 3x \), height \( = x^{3} \).

Step2: Multiply the expressions

Multiply the three terms: \( V=(2x^{2})\times(3x)\times(x^{3}) \). Using the rule of exponents \( a^{m}\times a^{n}=a^{m + n} \) and multiplying the coefficients: \( 2\times3 = 6 \), and for the variables \( x^{2}\times x\times x^{3}=x^{2 + 1+ 3}=x^{6} \). So, \( V = 6x^{6} \).

Step3: Substitute \( x = 3 \) into the formula

When \( x = 3 \, \text{cm} \), substitute into \( V = 6x^{6} \). Calculate \( x^{6}=3^{6}=729 \). Then \( V = 6\times729 \).

Step4: Compute the numerical value

\( 6\times729 = 4374 \, \text{cm}^3 \).

Answer:

The volume expression is \( 6x^{6} \) and the actual volume when \( x = 3 \, \text{cm} \) is \( 4374 \, \text{cm}^3 \).