Select the correct answer.
Wanda is a cake designer with a specialty in rectangular silk screen photo cakes. For every cake that she makes, the width of the cake is 4 inches more than the width of the photo in the center of the cake, and the length of every cake is two times its width. The area of the cake Wanda is currently working on is at least 254 square inches.
If [tex]$x$[/tex] represents the width of the photo, which inequality represents this situation?
A. [tex]$2 x^2+16 x+32 \geq 254$[/tex]
B. [tex]$x^2+8 x+16 \geq 254$[/tex]
C. [tex]$8 x^2+64 x+128 \geq 254$[/tex]
D. [tex]$x^2+4 x \geq 254$[/tex]
Answer (1)
Define the width of the photo as x .
Express the width and length of the cake in terms of x : width = x + 4 , length = 2 ( x + 4 ) .
Formulate the inequality for the area of the cake: ( x + 4 ) × 2 ( x + 4 ) \tIeq 254 .
Simplify the inequality to match one of the given options: 2 x 2 + 16 x + 32 \tIeq 254 .
The correct answer is A. 2 x 2 + 16 x + 32 ≥ 254 .
Explanation
Understanding the Problem Let's break down this problem step by step to find the correct inequality. We're given that the width of the cake is 4 inches more than the width of the photo, and the length of the cake is twice its width. The area of the cake is at least 254 square inches. We need to translate this information into an inequality using x to represent the width of the photo.
Expressing Cake Dimensions Let x be the width of the photo. Then, the width of the cake is x + 4 inches. Since the length of the cake is two times its width, the length of the cake is 2 ( x + 4 ) inches.
Formulating the Inequality The area of a rectangle is given by the formula: Area = width × length. In this case, the area of the cake is ( x + 4 ) × 2 ( x + 4 ) . We are told that the area is at least 254 square inches, which means the area is greater than or equal to 254. Therefore, we can write the inequality as: ( x + 4 ) × 2 ( x + 4 ) \tIeq 254
Simplifying the Inequality Now, let's simplify the inequality:
First, we can rewrite the left side as: 2 ( x + 4 ) ( x + 4 ) = 2 ( x + 4 ) 2 Expanding ( x + 4 ) 2 , we get: ( x + 4 ) 2 = x 2 + 8 x + 16 Now, multiply by 2: 2 ( x 2 + 8 x + 16 ) = 2 x 2 + 16 x + 32 So, the inequality becomes: 2 x 2 + 16 x + 32 \tIeq 254
Identifying the Correct Option Comparing our simplified inequality 2 x 2 + 16 x + 32 \tIeq 254 with the given options, we see that it matches option A.
Final Answer Therefore, the correct inequality that represents this situation is: 2 x 2 + 16 x + 32 \tIeq 254
Examples
Imagine you're designing a rectangular garden with a path around a central flower bed. The flower bed's width is 'x' feet. The garden's width is 'x + 4' feet (to accommodate a 2-foot path on each side), and the garden's length is twice its width, or '2(x + 4)' feet. If you want the garden's area to be at least 254 square feet, you can use the inequality 2 x 2 + 16 x + 32 ≥ 254 to determine the minimum size 'x' of the flower bed. This ensures your garden meets your space requirements while maintaining the desired path width.
