Problem 84 - Algebra and Trigonometry Real Mathematics, Real People

Question

A rectangular region of length \(x\) and width \(y\) has an area of 500 square meters. (a) Write the width \(y\) as a function of \(x .\) (b) Determine the domain of the function based on the physical constraints of the problem. (c) Sketch a graph of the function and determine the width of the rectangle when \(x=30\) meters.

Step-by-Step Solution

Answer

The width 'y' as a function of 'x' is \(y= \frac{500}{x}\), the domain of the function is \(0 < x < \infty\), and when \(x=30\) meters, the width 'y' is 16.7 meters.

Step 1: (a) Write the width 'y' as a function of 'x'

Given that the area of the rectangle 'A' is 500 square meters and area is given by the product of length 'x' and width 'y', then this relationship can be expressed as the equation \(A = x \cdot y\). Solving this equation for 'y' to express 'y' as a function of 'x' gives \(y = \frac{A}{x}\). Substituting 500 for the area 'A', the function becomes \(y = \frac{500}{x}\).

Step 2: (b) Determine the domain of the function

The domain of a function are the values of 'x' for which the function is defined. For this scenario, 'x' represents the length of a rectangle which has to be greater than 0 and cannot be infinite, because we cannot have a rectangle of zero or infinite length. However, when 'x' becomes too large (approaching to infinity), then 'y' becomes very small (approaching to zero) which is not realistic scenario for a rectangle. Therefore, the physical constraints of the problem indicate that the domain of this function is \(0 < x < \infty\). In other words, 'x' can be any positive real number, but not zero or infinity.

Step 3: (c) Sketch a graph of the function and determine the width of the rectangle when \(x=30\) meters

The function \(y = \frac{500}{x}\) is a reciprocal function and its graph is a hyperbola that approaches the x-axis (y = 0) as 'x' increases and the y-axis (x = 0) as 'y' increases. A sketch of this function would decline as 'x' increases after 0. However, note that the graph would no exist at x = 0 due to the physical constraints mentioned in previous step. To determine the width 'y', when \(x = 30\) meters, insert '30' into the function for 'x': \(y = \frac{500}{30} = 16.7\) meters.