Draw a diagram representing the situation. How can you express the area of the walkway as a difference of areas?
Width: 15 ft Length: 45 ft
Practice makes perfect
We want to find the dimensions of the pavilion. To do so, let's use the given bullet point questions.
Diagram
In most cases, when dealing with a geometry exercise, the first thing to do is to draw a diagram. It helps to visualize the problem then to find the solution. First, let's draw the rectangular pavilion. Its width is x and its length is three times the width, 3x.
Next, we will draw the walkway. It is 3ft wide around the pavilion.
Walkway Area Expression
The walkway is equivalent to the shaded region on the diagram.
The shaded region is the bigger rectangle excluding the smaller rectangle. Therefore, we can express the area of the walkway as the area of the bigger rectangle minus the area of the smaller rectangle.
A=A_b-A_s
Here, A represents the area of the walkway, A_b is the area of the bigger rectangle, and A_s is the area of the smaller rectangle. Now we have to calculate the areas of the bigger rectangle and the smaller rectangle. Recall the formula for the area of a rectangle.
A=l w
In this formula, l is the length of the rectangle and w is the width of the rectangle. Let's start by calculating the area of the bigger rectangle. We have to determine its dimensions first.
The length of the bigger rectangle is 3+3x+3, and its width is 3+x+3.
l &= 3+3x+3=3x+6
w &= 3+x+3=x+6
Let's substitute these values into the formula for the area of a rectangle and simplify!
Recall that the area of the walkway A is equal to the difference of A_b and A_s.
A=A_b-A_s=3x^2+24x+36-3x^2=24x+36
Pavilion Dimensions
We have found the area of the walkway, depending on the value of x.
A=24x+36
To cover the walkway, we want to use all stones. There is enough stones to cover 396ft^2. Therefore, the area of the walkway must be equal to the area that can be covered with the stones.
24x+36=396
Let's solve this equation for x.
Recall that x represents the width of the pavilion. We have that the pavilion should be 15ft wide. The length of the pavilion is three times its width, so the pavilion should be 45ft long.
