Final answer:
To determine the dimensions of the rectangular playground, we set up a quadratic equation 100 = L * (L + 15) based on the given area and width's relation to the length. By factoring the equation, we found the length of the playground to be 5 yards and the width to be 20 yards.
Step-by-step explanation:
To determine the length and width of the playground using factors, we need to set up an equation based on the information provided in the question. We know the area (A) of the rectangle is given as A = 100 square yards. Let's denote the length of the playground as L and the width as W. According to the problem, the width is 15 yards longer than its length, so we can write W = L + 15.
Since the area of a rectangle is calculated by multiplying the length by the width (A = L × W), we have:
A = L × (L + 15). Replacing A with 100 square yards, we get 100 = L × (L + 15). To solve this quadratic equation, let's first expand it:
100 = L^2 + 15L.
Next, we bring all terms to one side of the equation to set it equal to zero:
L^2 + 15L - 100 = 0.
Now we need to factor this quadratic equation. Factors of -100 that add up to 15 are 20 and -5. Therefore, we can rewrite the equation as:
(L + 20)(L - 5) = 0.
Setting each factor equal to zero gives us:
L + 20 = 0 or L - 5 = 0,
which results in L = -20 or L = 5. Since a length cannot be negative, the actual length of the playground is 5 yards. The actual width, using W = L + 15, is then 5 + 15 = 20 yards.