Question

A prism's volume is given by the expression
6k2 – 13k + 5. The area of the base of the
prism is 2k – 1.



Which expression represents the height of the prism?

k + 4
3k – 5
3k – 8 +
4k – 7 –

Answer 1

To find the height of the prism, we can use the formula for the volume of a prism, which is given by:

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

From your question, we know the volume of the prism is \(6k^2 - 13k + 5\) and the area of the base is \(2k - 1\). We are asked to find the height of the prism, which we can denote as \(h\).

We can set up the equation:

\[
6k^2 - 13k + 5 = (2k - 1) \cdot h
\]

To find \(h\), we'll divide the polynomial expression \(6k^2 - 13k + 5\) by the binomial \(2k - 1\).

Performing polynomial long division:

1. Divide the leading term of the numerator \(6k^2\) by the leading term of the denominator \(2k\) to get \(3k\).
2. Multiply \(3k\) by the entire denominator \(2k - 1\):

\[
3k(2k - 1) = 6k^2 - 3k
\]

3. Subtract this product from the original polynomial:

\[
(6k^2 - 13k + 5) - (6k^2 - 3k) = -10k + 5
\]

4. Now take the new leading term \(-10k\) and divide it by the leading term of the denominator \(2k\):

\[
-10k / 2k = -5
\]

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

\[
-5(2k - 1) = -10k + 5
\]

6. Subtract this from \(-10k + 5\):

\[
(-10k + 5) - (-10k + 5) = 0
\]

Thus, the division is exact, and we have:

\[
h = 3k - 5
\]

So, the expression that represents the height of the prism is:

\[
\boxed{3k - 5}
\]