Question

Explain which operation to use to solve the following problem. Then, solve the problem.


The volume of a cylinder is represented by the polynomial 2x3+5x2+3x+18
. The height of the cylinder is represented by the polynomial x+3
. What expression represents the area of the base of the cylinder?

(2 points)

Answer 1

To find the area of the base of a cylinder given the volume and height, you can use the formula for the volume of a cylinder:

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

From the problem, we know the volume of the cylinder is represented by the polynomial:

\[
V = 2x^3 + 5x^2 + 3x + 18
\]

And the height of the cylinder is represented by the polynomial:

\[
h = x + 3
\]

To find the area of the base, we need to solve for the base area (B) using the formula:

\[
B = \frac{V}{h}
\]

We will divide the polynomial representing the volume \( V \) by the polynomial representing the height \( h \):

\[
B = \frac{2x^3 + 5x^2 + 3x + 18}{x + 3}
\]

Now, we will perform polynomial long division.

1. Divide the first term of the numerator by the first term of the denominator:

\[
\frac{2x^3}{x} = 2x^2
\]

2. Multiply \( 2x^2 \) by the entire denominator \( x + 3 \):

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

3. Subtract this result from the original polynomial:

\[
(2x^3 + 5x^2 + 3x + 18) - (2x^3 + 6x^2) = -x^2 + 3x + 18
\]

4. Now, divide the first term of the new polynomial by the first term of the denominator again:

\[
\frac{-x^2}{x} = -x
\]

5. Multiply \(-x\) by the entire denominator:

\[
-x(x + 3) = -x^2 - 3x
\]

6. Subtract this from the result we got in the previous step:

\[
(-x^2 + 3x + 18) - (-x^2 - 3x) = 6x + 18
\]

7. Now, divide the first term of the new polynomial \( 6x + 18 \) by the first term of the denominator:

\[
\frac{6x}{x} = 6
\]

8. Multiply \( 6 \) by the entire denominator:

\[
6(x + 3) = 6x + 18
\]

9. Subtract this from \( 6x + 18 \):

\[
(6x + 18) - (6x + 18) = 0
\]

Thus, there is no remainder. The entire expression simplifies to:

\[
B = 2x^2 - x + 6
\]

So, the expression that represents the area of the base of the cylinder is:

\[
\boxed{2x^2 - x + 6}
\]