To find the height of the shipping box given the volume function \( f(x) \) and the functions for length \( g(x) \) and width \( h(x) \), we can use the relationship:
\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \]
This means the height function \( k(x) \) can be expressed as:
\[ k(x) = \frac{f(x)}{g(x) \cdot h(x)} \]
Let's compute \( g(x) \cdot h(x) \) first:
\[ g(x) = x + 3 \] \[ h(x) = x - 2 \] \[ g(x) \cdot h(x) = (x + 3)(x - 2) = x^2 + x - 6 \]
Now, substitute \( g(x) \cdot h(x) \) into the height formula:
\[ k(x) = \frac{f(x)}{g(x) \cdot h(x)} = \frac{2x^3 + 3x^2 - 11x - 6}{x^2 + x - 6} \]
Next, we need to find the domain. The domain of \( k(x) \) excludes values of \( x \) that make the denominator zero. We solve for \( x \) in:
\[ x^2 + x - 6 = 0 \]
Factoring gives:
\[ (x - 2)(x + 3) = 0 \]
So, \( x = 2 \) and \( x = -3 \) are the values to exclude. Thus, the domain is:
\[ (-\infty, -3) \cup (-3, 2) \cup (2, \infty) \]
Now, reviewing the possible answers provided, we can clarify that:
- The height function \( k(x) \) is \( \frac{2x^3 + 3x^2 - 11x - 6}{(x + 3)(x - 2)} \).
- The domain excludes \( x = -3 \) and \( x = 2 \).
Therefore, the height function \( k(x) \) does not match any of the responses given regarding the representation in the form \( (f \cdot g \cdot h)(x) \), nor does it accurately reflect the domain specified in any of the provided options.
So to summarize:
- The function representing the height is \( \frac{f(x)}{g(x) \cdot h(x)} \).
- The domain is \( (-\infty, -3) \cup (-3, 2) \cup (2, \infty) \).
1 answer