Answer:
below
Step-by-step explanation:
To solve this problem, we can use the formula for the volume of a rectangular box:
\[ V = l \times w \times h \]
Given that the volume \( V \) varies directly with the length \( l \) and the proportionality constant \( k = 24 \), we can express this relationship as:
\[ V = k \times l \]
Now, we can rearrange this equation to solve for the length \( l \):
\[ l = \frac{V}{k} \]
Since the girth of the box is the perimeter formed by the width and height, and the width is not given, let's denote the width as \( w \). Then, the girth \( G \) can be expressed as:
\[ G = 2w + 2h \]
Given that the girth \( G \) is 20 inches, we can substitute this into the equation:
\[ 20 = 2w + 2h \]
\[ 10 = w + h \]
Now, let's express the width \( w \) in terms of the length \( l \) using the proportionality constant \( k \):
\[ w = \frac{V}{k \times h} \]
Substitute the expression for \( w \) into the equation for the girth:
\[ 10 = \frac{V}{k \times h} + h \]
\[ 10 = \frac{V + kh^2}{k \times h} \]
Now, we can solve for \( h \):
\[ 10k \times h = V + kh^2 \]
\[ 10k \times h - kh^2 = V \]
\[ h(10k - kh) = V \]
\[ h(10 - k) = V \]
Given \( k = 24 \), plug in the value:
\[ h(10 - 24) = V \]
\[ -14h = V \]
\[ h = -\frac{V}{14} \]
Now, substitute the given value of \( k = 24 \) and \( G = 20 \) into the equation to find the height \( h \). However, it's crucial to note that a negative height doesn't make sense in this context. Let's reassess the problem.
