Question

The volume of a rectangular shipping box is given by the function v(x)=2x3+x2−25x+12
. The width of the shipping box is given by the function w(x)=x−3
and the height of the box is given by the function h(x)=x+4
. Which function represents the length of the shipping box? (1 point)
Responses

l(x)=2x2−1
l times x is equal to 2 x squared minus 1

l(x)=2x
l times x is equal to 2 x

l(x)=2x−1
l times x is equal to 2 x minus 1

l(x)=2x+1

Answer 1

To find the function that represents the length of the shipping box, we can use the relationship between the volume \( v \), width \( w \), height \( h \), and length \( l \) of the rectangular box, given by:

\[
v(x) = l(x) \cdot w(x) \cdot h(x)
\]

From the problem, we have:

- Volume function: \( v(x) = 2x^3 + x^2 - 25x + 12 \)
- Width function: \( w(x) = x - 3 \)
- Height function: \( h(x) = x + 4 \)

We can express the length function in terms of the volume, width, and height:

\[
l(x) = \frac{v(x)}{w(x) \cdot h(x)}
\]

Now, we can calculate \( w(x) \cdot h(x) \):

\[
w(x) \cdot h(x) = (x - 3)(x + 4)
\]

Using the distributive property:

\[
w(x) \cdot h(x) = x^2 + 4x - 3x - 12 = x^2 + x - 12
\]

Next, we will find \( l(x) \):

\[
l(x) = \frac{2x^3 + x^2 - 25x + 12}{x^2 + x - 12}
\]

Now we need to perform polynomial long division to divide \( 2x^3 + x^2 - 25x + 12 \) by \( x^2 + x - 12 \).

1. Divide \( 2x^3 \) by \( x^2 \) to get \( 2x \).
2. Multiply \( 2x \) by \( x^2 + x - 12 \) to get \( 2x^3 + 2x^2 - 24x \).
3. Subtract \( (2x^3 + 2x^2 - 24x) \) from \( (2x^3 + x^2 - 25x + 12) \):

\[
(2x^3 + x^2 - 25x + 12) - (2x^3 + 2x^2 - 24x) = (x^2 - 2x^2) + (-25x + 24x) + 12 = -x^2 - x + 12
\]

4. Now, divide \( -x^2 \) by \( x^2 \) to get \( -1 \).
5. Multiply \( -1 \) by \( x^2 + x - 12 \) to get \( -x^2 - x + 12 \).
6. Subtract:

\[
(-x^2 - x + 12) - (-x^2 - x + 12) = 0
\]

So, we can conclude:

\[
l(x) = 2x - 1
\]

Thus, the function that represents the length of the shipping box is:

\[
\boxed{l(x) = 2x - 1}
\]