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 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!
A_b=l w
A_b=( 3x+6)( x+6)
A_b=3x(x+6)+6(x+6)
A_b=3x^2+18x+6x+36
A_b=3x^2+24x+36
Next, we will calculate the area of the smaller rectangle, or the pavilion. The length of this rectangle is 3x and its width is x.
A_s=l w
A_s= 3x( x)
A_s=3x^2
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.
