Answer:
The dimensions of the rectangle that maximizes the area is 24 inches by 24 inches.
Step-by-step explanation:
We are given that a metallic bar with a length of 96 inches was used to make a rectangular frame.
And we want to find the dimensions of the frame that yields the maximum area.
Since the bar has a length of 96 inches, the perimeter of the rectangle must total 96. Let <em>l</em> represent the length of the rectangle and <em>w</em> its width. Hence:
![96 = 2\left( \ell + w \right)](https://tex.z-dn.net/?f=96%20%3D%202%5Cleft%28%20%5Cell%20%2B%20w%20%5Cright%29)
Divide:
![\displaystyle \ell + w = 48](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cell%20%2B%20w%20%3D%2048)
The area of the rectangle will be:
![\displaystyle A = w\ell](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%20%3D%20w%5Cell)
From the first equation, solve for either variable:
![\displaystyle w = 48 -\ell](https://tex.z-dn.net/?f=%5Cdisplaystyle%20w%20%3D%2048%20-%5Cell)
Substitute:
![\displaystyle A = \ell (48 -\ell)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%20%3D%20%5Cell%20%2848%20-%5Cell%29)
Distribute:
![\displaystyle A = 48\ell - \ell^2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%20%3D%2048%5Cell%20-%20%5Cell%5E2)
Since the area of the rectangle is represented with a quadratic equation with a negative leading coefficient, the maximum area will occur at its vertex point.
The vertex of a quadratic is given by:
![\displaystyle \text{Vertex} = \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ctext%7BVertex%7D%20%3D%20%5Cleft%28-%5Cfrac%7Bb%7D%7B2a%7D%2C%20f%5Cleft%28-%5Cfrac%7Bb%7D%7B2a%7D%5Cright%29%5Cright%29)
In this case, <em>a</em> = -1<em>, b</em> = 48, and <em>c</em> = 0.
Then the length for which the maximum area occurs is:
![\displaystyle \ell = -\frac{(48)}{2(-1)} = 24](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cell%20%3D%20-%5Cfrac%7B%2848%29%7D%7B2%28-1%29%7D%20%3D%2024)
And the width will be:
![\displaystyle w = 48 - \left(24\right) = 24](https://tex.z-dn.net/?f=%5Cdisplaystyle%20w%20%3D%2048%20-%20%5Cleft%2824%5Cright%29%20%3D%2024)
In conclusion, the dimensions of the rectangle that maximizes the area is 24 inches by 24 inches.