Question

1. marias garden has a rectangular shape with a length that is 3 feet longer than twice the width. if the area of the garden is 65 square feet, then what are the dimensions of the garden?

Explanation:

Step1: Define variables for width

Let width = $w$

Step2: Express length in terms of $w$

Length = $2w + 3$

Step3: Set up area equation

Area = length $\times$ width, so:
$$w(2w + 3) = 65$$

Step4: Expand and rearrange to quadratic

$$2w^2 + 3w - 65 = 0$$

Step5: Solve quadratic equation

Use quadratic formula $w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a=2$, $b=3$, $c=-65$:
$$w = \frac{-3 \pm \sqrt{3^2 - 4(2)(-65)}}{2(2)}$$
$$w = \frac{-3 \pm \sqrt{9 + 520}}{4}$$
$$w = \frac{-3 \pm \sqrt{529}}{4}$$
$$w = \frac{-3 \pm 23}{4}$$
Take positive root (width can't be negative):
$$w = \frac{-3 + 23}{4} = \frac{20}{4} = 5$$

Step6: Calculate length

Length = $2(5) + 3 = 13$

Answer:

The width of the garden is 5 feet, and the length is 13 feet.