Problem Statement:
A rectangular garden has a length that is 3 meters longer than its width. If the area of the garden is 54 square meters, what are the dimensions of the garden?
---
Solution:
1.
Define Variables:
Let the width of the garden be denoted by \( w \) (in meters).
Since the length is 3 meters longer than the width, the length can be expressed as \( w + 3 \) (in meters).
2.
Area Formula:
The area of a rectangle is given by the formula:
\[
\text{Area} = \text{length} \times \text{width}
\]
According to the problem, the area of the garden is 54 square meters. Therefore, we can write:
\[
(w + 3) \cdot w = 54
\]
3.
Formulate the Equation:
Expanding the left-hand side of the equation:
\[
w^2 + 3w = 54
\]
Rearrange the equation to form a standard quadratic equation:
\[
w^2 + 3w - 54 = 0
\]
4.
Solve the Quadratic Equation:
We solve the quadratic equation \( w^2 + 3w - 54 = 0 \) using the quadratic formula:
\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1 \), \( b = 3 \), and \( c = -54 \). Substituting these values into the formula:
\[
w = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-54)}}{2 \cdot 1}
\]
Simplify inside the square root:
\[
w = \frac{-3 \pm \sqrt{9 + 216}}{2}
\]
\[
w = \frac{-3 \pm \sqrt{225}}{2}
\]
Since \( \sqrt{225} = 15 \), we have:
\[
w = \frac{-3 \pm 15}{2}
\]
5.
Find the Possible Solutions:
There are two possible solutions for \( w \):
\[
w = \frac{-3 + 15}{2} = \frac{12}{2} = 6
\]
\[
w = \frac{-3 - 15}{2} = \frac{-18}{2} = -9
\]
Since the width cannot be negative, we discard \( w = -9 \). Thus, the width is:
\[
w = 6
\]
6.
Determine the Length:
The length is given by \( w + 3 \):
\[
\text{Length} = 6 + 3 = 9
\]
7.
Verify the Solution:
Check the area with the calculated dimensions:
\[
\text{Area} = \text{length} \times \text{width} = 9 \times 6 = 54
\]
The area matches the given value, confirming the solution is correct.
---
Final Answer:
The dimensions of the garden are:
\[
\boxed{6 \text{ meters (width)} \text{ and } 9 \text{ meters (length)}}
\]
Parent Tip: Review the logic above to help your child master the concept of things fall apart worksheet.
