Problem 149 - College Algebra Essentials

Question

A pool measuring 10 meters by 20 meters is surrounded by a path of uniform width, as shown in the figure at the top of the next column. If the area of the pool and the path combined is 600 square meters, what is the width of the path?

Step-by-Step Solution

Answer

The width of the path is approximately 2.77 meters.

Step 1: Calculate the rea of the pool

The area A of a rectangle is given by the product of its length (L) and breadth (B), i.e., A = L × B. The pool is rectangular with length 20 meters and breadth 10 meters, so its area is \( A = 20\,m \times 10\,m = 200\,m^2 \).

Step 2: Calculate the area of the path

We know that the combined area of the pool and the path is 600 square meters. If we subtract the area of the pool from this total area, we get the area of the path alone. Therefore, the area of the path alone is \( 600\,m^2 - 200\,m^2 = 400\,m^2 \).

Step 3: Compute the width of the path

The path surrounds the pool, thus forming a larger rectangle. Hence, the total length and breadth of the combined figure (pool and path) would be \((20 + 2x)\,m\) and \((10 + 2x)\,m\), respectively, where x is the width of the path. Therefore, the total area of the pool and the pathway together should be equal to the area of the larger rectangle. That is, we can write the equation: \((20 + 2x) \times (10 + 2x) = 600\). We simplify this equation to get a quadratic equation which we can solve to find the value for x. The equation is: \(4x^2 + 60x - 400 = 0\). Using the quadratic formula, we can solve for \(x\).

Step 4: Solve the Quadratic Equation

The quadratic formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substituting \(a = 4\), \(b = 60\), and \(c = -400\) into the formula, we can solve for \(x\), which is the width of the pathway. The answer for \(x\) should be positive because it is not possible to have a negative length in this context.