Question

a rectangular courtyard has an area of 104 square yards. the length of the playground is 3 yards shorter than twice the width. find the length of the playground. 13 yards 8 yards 10 yards 15 yards

Explanation:

Step1: Define variables

Let the width of the courtyard be $w$ yards. Then the length $l = 2w - 3$ yards.

Step2: Set up area - equation

The area of a rectangle is $A=l\times w$. Given $A = 104$ square yards, we substitute $l$ and $A$ into the formula: $(2w - 3)w=104$.

Step3: Expand the equation

Expand $(2w - 3)w$ to get $2w^{2}-3w = 104$, which can be rewritten as $2w^{2}-3w - 104 = 0$.

Step4: Solve the quadratic equation

For a quadratic equation $ax^{2}+bx + c = 0$ (here $a = 2$, $b=-3$, $c = - 104$), we can use the quadratic formula $w=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ or factor. Factoring $2w^{2}-3w - 104$ gives $(2w + 13)(w - 8)=0$.
Setting each factor equal to zero:

• $2w+13 = 0$ gives $w=-\frac{13}{2}$, but width cannot be negative, so we discard this solution.
• $w - 8=0$ gives $w = 8$ yards.

Step5: Find the length

Since $l = 2w-3$ and $w = 8$, then $l=2\times8 - 3=16 - 3=13$ yards.

Answer:

13 yards