Charice is planning the lines for her own basketball court. The free throw section will be a rectangle with a width that will be 2.25 meters greater than the width. Using 3.14 for... Charice is planning the lines for her own basketball court. The free throw section will be a rectangle with a width that will be 2.25 meters greater than the width. Using 3.14 for pi, determine the dimensions of the free throw section. Read more

Steps to Solve

1. Define Variables Let the width of the free throw section be represented as ( w ) meters. Since the length is 2.25 meters greater than the width, we can express the length as ( l = w + 2.25 ).

2. Set Up the Area Equation The area ( A ) of the rectangle (the free throw section) is given as ( A = l \cdot w ).

Substituting the expression for length into the area equation gives: $$ A = (w + 2.25) \cdot w $$

3. Substitute the Given Area We know the area of the free throw section is 3.128 m². Therefore, we set up the equation: $$ 3.128 = (w + 2.25) \cdot w $$

4. Rearrange the Equation Expanding the equation leads us to: $$ 3.128 = w^2 + 2.25w $$

5. Set the Equation to Zero Rearranging gives us a standard quadratic equation: $$ w^2 + 2.25w - 3.128 = 0 $$

6. Use the Quadratic Formula To solve for ( w ), we apply the quadratic formula: $$ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ Here, ( a = 1 ), ( b = 2.25 ), and ( c = -3.128 ).

7. Calculate the Discriminant First, compute ( b^2 - 4ac ): $$ b^2 - 4ac = (2.25)^2 - 4(1)(-3.128) $$

8. Find the Value of ( w ) Substituting back into the quadratic formula gives: $$ w = \frac{-2.25 \pm \sqrt{value}}{2} $$ Calculate the two potential values for ( w ) and keep the one that makes sense (positive).

9. Determine Length ( l ) Once ( w ) is found, the length ( l ) is computed as: $$ l = w + 2.25 $$