The perimeter of a rectangular soccer field is 20/3 meters. If the width of the soccer field is the reciprocal of the length, find the dimensions of the soccer field.
bogus! what soccer field is that small?
However, ignoring real-world considerations,
2(x + 1/x) = 20/3
x = 3 or 1/3
To find the dimensions of the soccer field, we'll use the given information about the perimeter and the relationship between the width and length.
Let's assume:
Length of the soccer field = L meters
Width of the soccer field = W meters
We're told that the width is the reciprocal of the length. In mathematical terms, this can be expressed as:
W = 1/L
The formula to calculate the perimeter of a rectangle is:
Perimeter = 2 * (Length + Width)
Given that the perimeter is 20/3 meters, we can set up the equation:
20/3 = 2 * (L + W)
Substituting the value of W from the given relation:
20/3 = 2 * (L + 1/L)
Now, let's simplify this equation to solve for L:
Multiply both sides by 3 to get rid of the fraction:
20 = 6 * (L + 1/L)
Divide both sides by 6 to isolate the expression:
20/6 = L + 1/L
Simplifying the left side:
10/3 = L + 1/L
To solve this quadratic equation, we'll multiply both sides by 3L to eliminate the fraction:
10L = 3L^2 + 3
Rearranging this equation to quadratic form:
3L^2 - 10L + 3 = 0
To factorize this quadratic equation, we need to find two numbers whose product is AC (3 * 3 = 9) and sum is B (-10).
The numbers are -1 and -9.
Rewriting the equation accordingly:
3L^2 - 9L - L + 3 = 0
Grouping terms and factoring:
(3L^2 - 9L) - (L - 3) = 0
3L(L - 3) - 1(L - 3) = 0
(3L - 1)(L - 3) = 0
Setting each factor equal to zero:
3L - 1 = 0
L - 3 = 0
Solving each equation separately:
3L = 1
L = 1/3
L = 3
Therefore, we have two possible values for L - 1/3 and 3.
If L = 1/3, substituting this value into the equation W = 1/L:
W = 1 / (1/3)
W = 3
So, the dimensions of the soccer field can be 1/3 meters (length) and 3 meters (width).
If L = 3, substituting this value into the equation W = 1/L:
W = 1 / 3
W = 1/3
So, the dimensions of the soccer field can also be 3 meters (length) and 1/3 meters (width).
Therefore, the possible dimensions for the soccer field are 1/3 meters by 3 meters or 3 meters by 1/3 meters.