A rectangular pan has a length that is [tex]$\frac{4}{3}$[/tex] the width. The total area of the pan is [tex]$432 in ^2$[/tex]. What is the width of the cake pan?
Answer (2)
Define the width as w , so the length is 3 4 w .
Express the area as A = lw = 3 4 w 2 .
Set up the equation 3 4 w 2 = 432 and solve for w 2 .
Take the square root to find the width: w = 324 = 18 . The width of the cake pan is 18 inches.
Explanation
Problem Analysis Let's analyze the problem. We are given that the length of a rectangular pan is 3 4 times its width, and the area of the pan is 432 square inches. We need to find the width of the cake pan.
Define Variables and Area Formula Let w be the width of the pan. Then the length l is 3 4 w . The area A of a rectangle is given by the formula A = lw .
Substitute Given Values We are given that the area A = 432 square inches. Substituting the expressions for l and w into the area formula, we get: A = lw = ( 3 4 w ) w = 3 4 w 2
Set up the Equation Now we can set up the equation: 3 4 w 2 = 432
Solve for w^2 To solve for w 2 , we multiply both sides of the equation by 4 3 :
w 2 = 432 × 4 3 = 324
Solve for w Now, we take the square root of both sides to find w :
w = 324 = 18
Final Answer Therefore, the width of the cake pan is 18 inches.
Examples
Understanding the area of rectangles is useful in many real-world scenarios. For example, if you're designing a garden and know the amount of space you have available (the area), and you want the length to be a certain proportion of the width, you can use this method to calculate the exact dimensions of your garden. Similarly, architects and interior designers use these calculations to plan room layouts and optimize space utilization.
The width of the cake pan is 18 inches. This is calculated by defining the width as w and finding the corresponding length based on the area. Using the formulas for area, we solve for width and find that it equals 18 inches.
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