Questions: Write a function to represent the display area in terms of (x). [ A(x)=36xy ] What dimensions should be used to maximize the display area? The dimensions should be (square) in by (square) in.

Write a function to represent the display area in terms of (x).
[ A(x)=36xy ]

What dimensions should be used to maximize the display area?

The dimensions should be (square) in by (square) in.

Transcript text: Write a function to represent the display area in terms of $x$. \[ A(x)=36 x y \] What dimensions should be used to maximize the display area? The dimensions should be $\square$ in by $\square$ in.

Solution

Solution Steps

Solution Approach

To maximize the display area, we need to find the critical points of the function $ A(x) = -x^2 + 18x $. This involves taking the derivative of $ A(x) $, setting it to zero, and solving for $ x $. Once we have $ x $, we can find the corresponding $ y $ value using the relationship between $ x $ and $ y $.

Step 1: Find the Critical Points

To maximize the display area $ A(x) = -x^2 + 18x $, we first find the derivative: \[ A'(x) = 18 - 2x \] Setting the derivative equal to zero to find critical points: \[ 18 - 2x = 0 \implies x = 9 \]

Step 2: Calculate the Corresponding $ y $ Value

Using the relationship between $ x $ and $ y $, we find $ y $ when $ x = 9 $: \[ y = \frac{36}{x} = \frac{36}{9} = 4 \]