/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 100 The length and width of a rectan... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 100

The length and width of a rectangular garden are 15 meters and 10 meters, respectively. A walkway of width \(x\) surrounds the garden. (a) Draw a diagram that gives a visual representation of the problem. (b) Write the equation for the perimeter \(y\) of the walkway in terms of \(x\) (c) Use a graphing utility to graph the equation for the perimeter. (d) Determine the slope of the graph in part (c). For each additional one- meter increase in the width of the walkway, determine the increase in its perimeter.

Short Answer

Expert verified
The equation for the perimeter of the walkaway in terms of \(x\) is \(y = 4x + 50\). The slope of the graph is 4, which means for each additional one-meter increase in the width of the walkway, the perimeter increases by 4 meters.

Step by step solution

01

Visual Representation

Create a diagram illustrating the rectangular garden with a walkway surrounding it. The rectangular garden is initially 15m by 10m, and the walkway is of width x meters.
02

Perimeter Equation

Understand the typical geometry rules: perimeter of rectangle is \(P = 2l + 2w\) (where \(l\) is length and \(w\) is width). The new length including the walkway becomes \(15+2x\) meters and the new width becomes \(10+2x\) meters. So the equation that governs the perimeter becomes: \(y = 2(15 + 2x) + 2(10 + 2x)\)
03

Reorganize Equation and Graph

Reorganize the equation obtained from Step 2 to : \(y = 4x + 50\). Use a graphing tool to graph this linear equation.
04

Determine the Slope

Since the equation of the perimeter is in slope-intercept form (\(y = mx + b\)), the coefficient of \(x\) is the slope. In this case, the slope is 4. This mean for every one-meter increment of the width of the walkway, the perimeter increases by 4 units.

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