Question

The area of a rectangle is (4m^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 given the area and the width, we can use the formula for the area of a rectangle:

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

Given that the area of the rectangle is \(4m^3 + 16m^2 - 23m - 15\) ft² and the width is \(\frac{m + 1}{2}\) ft, we can rearrange the formula to find the length:

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

Substituting in the given area and width:

\[
\text{Length} = \frac{4m^3 + 16m^2 - 23m - 15}{\frac{m + 1}{2}}
\]

To simplify this expression, we can multiply the numerator by the reciprocal of the width:

\[
\text{Length} = \left(4m^3 + 16m^2 - 23m - 15\right) \times \frac{2}{m + 1}
\]

Now we can simplify this by performing polynomial long division or factoring, if possible, to divide \(4m^3 + 16m^2 - 23m - 15\) by \(m + 1\).

Let's perform polynomial long division:

1. Divide the leading term of the numerator \(4m^3\) by the leading term of the denominator \(m\):
\[
4m^2
\]

2. Multiply \(4m^2\) by \(m + 1\):
\[
4m^3 + 4m^2
\]

3. Subtract this from the original polynomial:
\[
(4m^3 + 16m^2 - 23m - 15) - (4m^3 + 4m^2) = 12m^2 - 23m - 15
\]

4. Divide the leading term \(12m^2\) by \(m\):
\[
12m
\]

5. Multiply \(12m\) by \(m + 1\):
\[
12m^2 + 12m
\]

6. Subtract this from the previous remainder:
\[
(12m^2 - 23m - 15) - (12m^2 + 12m) = -35m - 15
\]

7. Divide the leading term \(-35m\) by \(m\):
\[
-35
\]

8. Multiply \(-35\) by \(m + 1\):
\[
-35m - 35
\]

9. Subtract this:
\[
(-35m - 15) - (-35m - 35) = 20
\]

Thus, the division gives us:

\[
4m^2 + 12m - 35 + \frac{20}{m + 1}
\]

Now substituting back to find the length:

\[
\text{Length} = 2(4m^2 + 12m - 35) + \frac{40}{m + 1}
\]

Simplifying further:

\[
\text{Length} = 8m^2 + 24m - 70 + \frac{40}{m + 1}
\]

The final answer for the length of the rectangle is:

\[
\text{Length} = 8m^2 + 24m - 70 + \frac{40}{m + 1} \text{ ft}
\]