Question

Lila is planning to build a rectangular garden in her backyard. She wants the length to be twice the width plus 5 meters. If the area of her garden is 250 square meters, then what is the width?

Answer 1

Let the width of the rectangular garden be \( w \) meters. According to the problem, the length \( l \) can be expressed in terms of the width as follows:

\[
l = 2w + 5
\]

The area \( A \) of a rectangle is given by the formula:

\[
A = l \times w
\]

From the problem, we know the area is 250 square meters:

\[
A = 250 \quad \text{square meters}
\]

Substituting the expression for the length into the area formula gives:

\[
w \times (2w + 5) = 250
\]

Now, we can expand this equation:

\[
2w^2 + 5w = 250
\]

Next, we will rearrange the equation to set it to zero:

\[
2w^2 + 5w - 250 = 0
\]

Now we can solve this quadratic equation using the quadratic formula:

\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In our case, \( a = 2 \), \( b = 5 \), and \( c = -250 \). First, we calculate the discriminant:

\[
b^2 - 4ac = 5^2 - 4 \times 2 \times (-250) = 25 + 2000 = 2025
\]

Next, we take the square root of the discriminant:

\[
\sqrt{2025} = 45
\]

Now we can apply the quadratic formula:

\[
w = \frac{-5 \pm 45}{2 \times 2}
\]

Calculating the two possible values for \( w \):

1. For the positive root:

\[
w = \frac{-5 + 45}{4} = \frac{40}{4} = 10
\]

2. For the negative root:

\[
w = \frac{-5 - 45}{4} = \frac{-50}{4} = -12.5 \quad \text{(not physically meaningful)}
\]

Thus, the only valid solution is:

\[
w = 10 \quad \text{meters}
\]

We should verify this solution. If \( w = 10 \):

\[
l = 2(10) + 5 = 20 + 5 = 25 \quad \text{meters}
\]

Now, let’s check the area:

\[
A = l \times w = 25 \times 10 = 250 \quad \text{square meters}
\]

Since our calculations are correct, we conclude that the width of Lila's garden is:

\[
\boxed{10} \quad \text{meters}
\]