Question

The area of a rectangle is (43m^3+16m^2-23m-15) ft^2. Find the length of the rectangle if the width is (m+1/2)

Answer 1

To find the length of the rectangle, we can use the formula for the area of a rectangle, which is given by:

\[
\text{Area} = \text{Length} \times \text{Width}
\]

In this case, the area is given as \( 43m^3 + 16m^2 - 23m - 15 \) ft², and the width is given as \( \left(m + \frac{1}{2}\right) \) ft.

We can find the length (L) by rearranging the area formula:

\[
L = \frac{\text{Area}}{\text{Width}}
\]

Substituting the area and the width into this formula, we have:

\[
L = \frac{43m^3 + 16m^2 - 23m - 15}{m + \frac{1}{2}}
\]

To perform the division, we can simplify \( m + \frac{1}{2} \) to a single fraction:

\[
L = \frac{43m^3 + 16m^2 - 23m - 15}{\frac{2m + 1}{2}} = \frac{2(43m^3 + 16m^2 - 23m - 15)}{2m + 1}
\]

Now we need to divide \( 43m^3 + 16m^2 - 23m - 15 \) by \( 2m + 1 \).

We can perform polynomial long division:

1. Divide the leading term of the numerator by the leading term of the divisor:
\[
\frac{43m^3}{2m} = \frac{43}{2}m^2
\]

2. Multiply \( \frac{43}{2}m^2 \) by \( 2m + 1 \):
\[
\frac{43}{2}m^2(2m + 1) = 43m^3 + \frac{43}{2}m^2
\]

3. Subtract this from the original polynomial:
\[
(43m^3 + 16m^2 - 23m - 15) - \left(43m^3 + \frac{43}{2}m^2\right) = \left(16m^2 - \frac{43}{2}m^2\right) - 23m - 15
\]

4. Combine like terms:
\[
16m^2 - \frac{43}{2}m^2 = \frac{32m^2 - 43m^2}{2} = \frac{-11m^2}{2}
\]
So, we have:
\[
-\frac{11}{2}m^2 - 23m - 15
\]

5. Repeat the process by taking the new leading term:
\[
\frac{-\frac{11}{2}m^2}{2m} = -\frac{11}{4}m
\]

6. Multiply:
\[
-\frac{11}{4}m(2m + 1) = -\frac{11}{2}m^2 - \frac{11}{4}m
\]

7. Subtract again:
\[
\left(-\frac{11}{2}m^2 - 23m - 15\right) - \left(-\frac{11}{2}m^2 - \frac{11}{4}m\right) = \left(-23 + \frac{11}{4}\right)m - 15 = -\frac{69}{4}m - 15
\]

8. Continue the process with:
\[
\frac{-\frac{69}{4}m}{2m} = -\frac{69}{8}
\]

9. Multiply:
\[
-\frac{69}{8}(2m + 1) = -\frac{69}{4}m - \frac{69}{8}
\]

10. Finally, subtract:
\[
-\frac{69}{4}m - 15 + \frac{69}{4}m + \frac{69}{8} = -15 + \frac{69}{8}
\]

Converting -15 to eighths:
\[
-15 = -\frac{120}{8}
\]
So the remainder becomes:
\[
-\frac{120}{8} + \frac{69}{8} = -\frac{51}{8}
\]

Putting this all together, we find:
\[
L = \frac{43}{2}m^2 - \frac{11}{4}m - \frac{69}{8}
\]

Thus, the final expression for the length \(L\) of the rectangle, given the width, is:

\[
L = \frac{2(43m^3 + 16m^2 - 23m - 15)}{2m + 1} = 43m^2 - 11m - \frac{69}{8}
\]

This polynomial represents the length of the rectangle in terms of \(m\).