Question

A rectangular patio has an area of 91 fl. The length is 6 feet

Answer 1

Answer:

Total area equation = tex]w(w+6)=91[/tex]

b/2 = 3

Dimensions of the patio: width = 7 feet, length = 13 feet

Step-by-step explanation:

The area of a rectangle is given the formula:

$A=wl$

where

$w$ is the width

$l$ is the length

We know from our problem that the area of the patio is 91 square feet, so $A=91$. We also know that the length is 6 feet longer then the width, so $l=w+6$.

Replacing values in our area equation

$A=wl$

$91=w(w+6)$

$w(w+6)=91$

Expanding the left side:

$w*w+6w=91$

$w^2+6w=91$

Remember that to complete the square we need to add half the coefficient of the linear term squared. The lineal term is $w$, so its coefficient is 6. Now, half its coefficient or $\frac{b}{2} =\frac{6}{2} =3$. Finally, $3^2=9$.

To complete the square we need to add 9 to both sides of the equation:

$w^2+6w+9=91+9$

$w^2+6w+9=100$

Notice that the left side is a perfect square trinomial (both $w^2$ and 9 are perfect squares), so we can express it as:

$(w+3)^2=100$

Now that we completed the square, we can solve our equation

- Take square root to both sides

$\sqrt{(w+3)^2} =\pm\sqrt{100}$

$w+3=\pm10$

- Subtract 3 from both results

$w=10-3,w=-10-3$

$w=7,w=13$

Since length cannot be negative, $w=7$ is the solution of our equation.

We now know that the width of our rectangular patio is 7 feet, so we can find its length:

$l=w+6$

$l=7+6$

$l=13$

We can conclude that half the coefficient of the width is $\frac{b}{2}=3$, the width of the patio is 7 feet, and its length is 13 feet.