Kaleab T. answered • 06/02/20
Math, science, test prep, and more!
Hey Cristiana,
Hope you're doing well!
We can set up a system of equations to solve this problem. Since we're dealing with a rectangle, let's call the length x and the width y. We know that the perimeter of the rectangle is 48 meters (from first sentence), and the area of the rectangle is 108 square meters (from second sentence). Perimeter is the sum of all four sides (x, x, y, y), and area is the product of the length and width (x*y), so we can use these two facts to set up our equations:
perimeter = 2x + 2y = 48
area = x*y = 108
Since we have two equations and two unknown variables, we can solve this system of equations to figure out what x and y are. Let's rearrange the area equation to isolate x:
x*y = 108
x = 108/y
Then replace the x in the perimeter equation by the expression we just found:
2x + 2y = 48
2(108/y) + 2y = 48
Now we can determine the value of y by multiplying everything by y and solving the quadratic equation:
2(108/y) + 2y = 48
216/y + 2y = 48
216 + 2y^2 = 48y
2y^2 - 48y + 216 = 0
y^2 - 24y + 108 = 0
(y-18)(y-6) = 0
So y can be 18 or 6. When we plug these in (one at a time) to the area equation to determine the value of x, we see that x can also be 18 or 6:
x*y = 108
x*18 = 108
x = 6
x*y = 108
x*6 =108
x = 18
So the two dimensions of the rectangle must be 18 and 6, since these two values yield a perimeter of 48 and area of 108. Since our units are meters, the dimensions of the garden are thus 18 meters and 6 meters.
Best,
Kaleab
