Question

The face of a rectangular solid block has an area represented by the polynomial A=2x2+6x+8
m², where x
is in meters. The height of the block is denoted as h=2x+3
m. What is the volume of the block in terms of x.
A.
B.
C.
D.

Answer 1

To find the volume of the rectangular solid block, we can use the formula for the volume \( V \) of a rectangular prism, which is given by:

\[
V = \text{Base Area} \times \text{Height}
\]

In this case, the base area \( A \) is represented by the polynomial:

\[
A = 2x^2 + 6x + 8 \, \text{m}^2
\]

And the height \( h \) is given by:

\[
h = 2x + 3 \, \text{m}
\]

Now, we can substitute these expressions into the volume formula:

\[
V = A \times h = (2x^2 + 6x + 8)(2x + 3)
\]

Next, we need to multiply these two expressions together. We will use the distributive property (also known as the FOIL method for polynomials) to expand this product:

\[
V = (2x^2 + 6x + 8)(2x + 3)
\]

Distributing each term in the first polynomial by each term in the second polynomial:

1. \( 2x^2 \cdot 2x = 4x^3 \)
2. \( 2x^2 \cdot 3 = 6x^2 \)
3. \( 6x \cdot 2x = 12x^2 \)
4. \( 6x \cdot 3 = 18x \)
5. \( 8 \cdot 2x = 16x \)
6. \( 8 \cdot 3 = 24 \)

Now, we sum these results:

\[
V = 4x^3 + (6x^2 + 12x^2) + (18x + 16x) + 24
\]

Combining like terms:

- The \( x^3 \) term: \( 4x^3 \)
- The \( x^2 \) terms: \( 6x^2 + 12x^2 = 18x^2 \)
- The \( x \) terms: \( 18x + 16x = 34x \)
- The constant: \( 24 \)

Putting it all together, we get:

\[
V = 4x^3 + 18x^2 + 34x + 24
\]

Thus, the volume of the block in terms of \( x \) is:

\[
\boxed{4x^3 + 18x^2 + 34x + 24}
\]