How do I set this up and solve?
You are planing a rectangular dining pavilion. Its length is 3 times its width x. You want a stone walkway that is 3 ft wide around the pavilion. You have enough stones to cover 396 ft squared and want to use them all in the walkway. What should the dimension of the pavilion be? How can you write a variable expression for the area of the walkway?
l=3w
area walkway: outside area -inside area
area walkaway=(l+6)x (w+6)-lw
=(3w+6)(w+6)-3w^2
solve for w. Then, solve for l
5w=42
How can you write a variable expression for the area of the walkway
To set up and solve this problem, follow these steps:
Step 1: Understand the problem
Read the problem carefully and identify the information given and what needs to be determined. In this case, the problem provides the following information:
- The length of the pavilion is 3 times its width (x).
- A stone walkway that is 3 ft wide will surround the pavilion.
- There are enough stones to cover an area of 396 ft² in the walkway.
The problem asks you to find the dimensions of the pavilion and write a variable expression for the area of the walkway.
Step 2: Assign variables
Identify the variables present in the problem. In this case, the variable x represents the width of the pavilion.
Step 3: Set up an equation for the area of the walkway
The area of the walkway can be calculated by subtracting the area of the pavilion from the combined area of the pavilion and the walkway. The area of the walkway can be expressed as follows:
Area of walkway = (length of pavilion + 6) * (width of pavilion + 6) - (length of pavilion * width of pavilion)
Step 4: Use the area of the walkway equation to solve for the width of the pavilion
Plug in the values from the problem into the equation and solve for the width of the pavilion. Here's what the equation looks like:
396 = (3x + 6) * (x + 6) - (3x * x)
Step 5: Simplify and solve the equation
Simplify the equation by multiplying the terms and combining like terms. Next, solve the equation by isolating the variable and simplifying further.
Once you have solved the equation, you will find the value of x, which represents the width of the pavilion. To determine the length of the pavilion, you can substitute this width value back into the equation given in the problem.