To find the dimensions of the sand volleyball court, we start by using the formulas for area and perimeter. Let the length of the court be l meters and the width be w meters.
The area of the court can be expressed as:
Area = l * w = 162
And the perimeter is given by:
Perimeter = 2(l + w) = 54
We can simplify the perimeter formula to:
l + w = 27
Now, we have a system of equations:
- l * w = 162
- l + w = 27
From the second equation, we can express one variable in terms of the other. Let’s solve for w:
w = 27 – l
Next, we substitute this expression for w into the area equation:
l * (27 – l) = 162
Expanding this gives:
27l – l2 = 162
Rearranging the equation results in a standard quadratic form:
l2 – 27l + 162 = 0
Now we can apply the quadratic formula:
l = (-b ± √(b² – 4ac)) / 2a
Here, a = 1, b = -27, and c = 162:
l = (27 ± √((-27)2 – 4 * 1 * 162)) / (2 * 1)
l = (27 ± √(729 – 648)) / 2
l = (27 ± √81) / 2
l = (27 ± 9) / 2
Now we calculate the two possible values for l:
1. l = (36) / 2 = 18
2. l = (18) / 2 = 9
Using l = 18, we can find w:
w = 27 – 18 = 9
Or with l = 9:
w = 27 – 9 = 18
Thus, the dimensions of the volleyball court are length = 18 meters and width = 9 meters.
In conclusion, the volleyball court is 18 meters long and 9 meters wide.