Modeling a Rectangular Pool Using Algebra: Length and Width Determination
Algebra is a powerful tool for solving problems in geometry, particularly when dealing with shapes that have specific relationships between their dimensions. This article will guide you through the process of determining the length and width of a rectangular pool, given certain conditions. We will break down the problem step by step using algebraic expressions and equations.
Understanding the Problem
Consider a rectangular pool where the length is 5 meters less than twice the width. If the perimeter of the pool is 74 meters, how can we determine the width and length of the pool?
Mathematical Representation
Let's define:
X Width of the pool (in meters) L Length of the pool (in meters)Given the problem conditions:
The relationship between the length and the width can be expressed as:
L 2X - 5.
The perimeter of the rectangle is defined as the sum of all its sides, which can be written as:
P 2L 2X 74.
Solving the Equations
Substitute the expression for L into the perimeter equation:
2(2X - 5) 2X 74.
Expanding this gives:
4X - 10 2X 74
Simplify the equation:
6X - 10 74
Add 10 to both sides to isolate the term with X:
6X 84
Solve for X by dividing both sides by 6:
X 14 meters.
Now, substitute X back into the expression for L:
L 2(14) - 5 23 meters.
Verification
Let's verify that our solution is correct by plugging the values back into the perimeter formula:
P 2L 2X 2(23) 2(14) 46 28 74 meters.
Generalizing the Problem
Suppose we have a rectangular pool with an area of 18 square meters. We will now explore how to use algebra to solve this problem as well.
Let L be the length and W be the width. The relationship is still the same as before:
L 2W - 5.
The area can be expressed as:
A LW.
Substitute the expression for L into the area equation:
18 (2W - 5)W.
Expanding this gives:
18 2W2 - 5W.
Rearrange to form a standard quadratic equation:
2W2 - 5W - 18 0.
Solve this quadratic equation using the quadratic formula:
W (frac{-b pm sqrt{b^2 - 4ac}}{2a}), where a 2, b -5, and c -18.
Calculate the discriminant:
(Delta b^2 - 4ac (-5)^2 - 4(2)(-18) 25 144 169.
Thus, W (frac{5 pm sqrt{169}}{4} frac{5 pm 13}{4}.)
This gives us two solutions:
W (frac{18}{4} 4.5) meters and W (frac{-8}{4} -2) meters (which is not physically meaningful in this context).
So, the width is 4.5 meters, and the length is:
L 2(4.5) - 5 4 meters.
Finally, verify the area:
A LW 4(4.5) 18 square meters, which confirms our solution.
Conclusion
In conclusion, determining the dimensions of a rectangular pool using algebra involves several steps. By defining the variables, establishing the relationships, and solving the resulting equations, we can find the exact dimensions of the pool. This process is applicable not only to pools but to any rectangular shape that we need to analyze in the context of geometry and algebra.