The problem asks us to find the dimensions of a rectangle, given:

1. The area of the rectangle is $52 \, \text{ft}^2$.
2. The length of the rectangle is $5 \, \text{ft}$ less than twice the width.

Step-by-step solution:

Step 1: Define variables

Let:

• $w$ = width of the rectangle (in feet),
• $l$ = length of the rectangle (in feet).

We are given two key relationships:

1. Area equation: $l \times w = 52$,
2. Length equation: $l = 2w - 5$.

Step 2: Substitute $l = 2w - 5$ into the area equation

From $l \times w = 52$, substitute $l = 2w - 5$: $(2w - 5) \cdot w = 52$

Simplify the equation: $2w^2 - 5w = 52$

Rearrange into standard quadratic form: $2w^2 - 5w - 52 = 0$

Step 3: Solve the quadratic equation

The quadratic equation is: $2w^2 - 5w - 52 = 0$

Use the quadratic formula: $w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here:

• $a = 2$,
• $b = -5$,
• $c = -52$.

Substitute into the formula: $w = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-52)}}{2(2)}$ $w = \frac{5 \pm \sqrt{25 + 416}}{4}$ $w = \frac{5 \pm \sqrt{441}}{4}$ $w = \frac{5 \pm 21}{4}$

Solve for the two possible values:

1. $w = \frac{5 + 21}{4} = \frac{26}{4} = 6.5$,
2. $w = \frac{5 - 21}{4} = \frac{-16}{4} = -4$ (not valid, since width cannot be negative).

Thus, $w = 6.5 \, \text{ft}$.

Step 4: Find $l$

Substitute $w = 6.5$ into $l = 2w - 5$: $l = 2(6.5) - 5 = 13 - 5 = 8 \, \text{ft}.$

Final Answer:

The dimensions of the rectangle are:

• Width: $6.5 \, \text{ft}$,
• Length: $8 \, \text{ft}$.

Would you like a detailed breakdown of any specific step?
Here are some related questions to expand this problem:

1. How do you determine the formula for the area of a rectangle?
2. What is the process for solving quadratic equations in general?
3. Can you explain why negative dimensions are not valid in this context?
4. How do we check if the solution satisfies both the area and length conditions?
5. How would the solution change if the area or length relationship was modified?

Tip: Always verify your solutions by substituting them back into the original equations to ensure accuracy!