{"answerArea": {"calculator": false, "chi2Table": false, "periodicTable": false, "tTable": false, "zTable": false}, "hints": [{"content": "The $x$-intercept is the point of intersection between the graph of the function $A(w)$ and the $w$-axis, which is the point on the graph where $A(w)=0$. In this case the horizontal axis represents the variable $w$, but it's common to call that axis the $x$-axis.\n\nThis graph has an $x$-intercept at $(20,0)$, which means that *when the width of the rectangle is $\\textit{20 cm}$, it has no area.*\n\n\n\n[[☃ image 1]]\n\n", "images": {}, "replace": false, "widgets": {"image 1": {"alignment": "block", "graded": true, "options": {"alt": "The first quadrant of a coordinate plane. The x-axis scales by two and is labeled w for the rectangle's width in centimeters. The y-axis scales by ten and is labeled A of w for the area of the rectangle. The graph of the function is a continuous curve. From left to right, it starts at zero, zero and increases through the point two, thirty-five, the point four, sixty-five, the point six, eighty-five, and the point eight, ninety until it reaches a local maximum at ten, one hundred. Then it decreases through the point twelve, ninety-five, the point fourteen, eighty-five, the point sixteen, sixty-five, and the point eighteen, thirty-five until it stops at the x-intercept twenty, zero. The point twenty, zero is plotted on the function.", "backgroundImage": {"height": 231, "url": "web+graphie:${☣ LOCALPATH}/images/2575f00f0975278c47adf72c41003d5d625d0a44", "width": 231}, "box": [231, 231], "caption": "", "labels": [], "range": [[0, 10], [0, 10]], "static": false, "title": ""}, "static": false, "type": "image", "version": {"major": 0, "minor": 0}}}}, {"content": "A relative maximum (or minimum) point is a point that is higher (or lower) than all of the points surrounding it.\n\nThis graph has a relative maximum point where $A(w)=100$, which means that *the greatest possible area of the rectangle is $\\textit{100 sq. cm}$.*\n\n\n\n[[☃ image 1]]\n\n", "images": {}, "replace": false, "widgets": {"image 1": {"alignment": "block", "graded": true, "options": {"alt": "The first quadrant of a coordinate plane. The x-axis scales by two and is labeled w for the rectangle's width in centimeters. The y-axis scales by ten and is labeled A of w for the area of the rectangle. The graph of the function is a continuous curve. From left to right, it starts at zero, zero and increases through the point two, thirty-five, the point four, sixty-five, the point six, eighty-five, and the point eight, ninety until it reaches a local maximum at ten, one hundred. Then it decreases through the point twelve, ninety-five, the point fourteen, eighty-five, the point sixteen, sixty-five, and the point eighteen, thirty-five until it stops at the x-intercept twenty, zero. The point ten, one hundred is plotted on the function.", "backgroundImage": {"height": 231, "url": "web+graphie:${☣ LOCALPATH}/images/68aaaa9e6a6611bd23128193d54321eaf4be53ea", "width": 231}, "box": [231, 231], "caption": "", "labels": [], "range": [[0, 10], [0, 10]], "static": false, "title": ""}, "static": false, "type": "image", "version": {"major": 0, "minor": 0}}}}, {"content": "An increasing (or decreasing) interval is a domain interval over which the function values increase (or decrease) as the input variable increases.\n\nIn this graph, the interval $[0,10]$ is an increasing interval. This means that *as the width of the rectangle is extended towards $\\textit{10 cm}$, the area of the rectangle grows.*\n\n\n\n[[☃ image 1]]\n\n", "images": {}, "replace": false, "widgets": {"image 1": {"alignment": "block", "graded": true, "options": {"alt": "The first quadrant of a coordinate plane. The x-axis scales by two and is labeled w for the rectangle's width in centimeters. The y-axis scales by ten and is labeled A of w for the area of the rectangle. The graph of the function is a continuous curve. From left to right, it starts at zero, zero and increases through the point two, thirty-five, the point four, sixty-five, the point six, eighty-five, and the point eight, ninety until it reaches a local maximum at ten, one hundred. Then it decreases through the point twelve, ninety-five, the point fourteen, eighty-five, the point sixteen, sixty-five, and the point eighteen, thirty-five until it stops at the x-intercept twenty, zero. The function is highlighted from x equals zero to x equals ten.", "backgroundImage": {"height": 231, "url": "web+graphie:${☣ LOCALPATH}/images/db7678c41480e307cfb2cd97360daae718f92503", "width": 231}, "box": [231, 231], "caption": "", "labels": [], "range": [[0, 10], [0, 10]], "static": false, "title": ""}, "static": false, "type": "image", "version": {"major": 0, "minor": 0}}}}, {"content": "To sum it up, the table should look as follows:\n\nFeature | Statement\n:-: | :-:\n$x$-intercept | When the width of the rectangle is $20\\text{ cm}$, it has no area.\nRelative maximum or minimum | The greatest possible area of the rectangle is $100\\text{ sq. cm}$.\nIncreasing or decreasing interval | As the width of the rectangle is extended towards $10\\text{ cm}$, the area of the rectangle grows.", "images": {}, "replace": false, "widgets": {}}], "itemDataVersion": {"major": 0, "minor": 1}, "question": {"content": "Vera made a rectangle using a string that is $40\\text{ cm}$ long.\n\n$A(w)$ models the area of the rectangle (in $\\text{sq. cm}$) as a function of the rectangle's width $w$ (in $\\text{cm}$).\n\n**Match each statement with the feature of the graph that most closely corresponds to it.**\n\n\n\n[[☃ image 1]]\n\n[[☃ matcher 1]]", "images": {}, "widgets": {"image 1": {"alignment": "block", "graded": true, "options": {"alt": "The first quadrant of a coordinate plane. The x-axis scales by one and is labeled w for the rectangle's width in centimeters. The y-axis scales by five and is labeled A of w for the area of the rectangle. The graph of the function is a continuous curve. From left to right, it starts at zero, zero and increases through the point two, thirty-five, the point four, sixty-five, the point six, eighty-five, and the point eight, ninety until it reaches a local maximum at ten, one hundred. Then it decreases through the point twelve, ninety-five, the point fourteen, eighty-five, the point sixteen, sixty-five, and the point eighteen, thirty-five until it stops at the x-intercept twenty, zero.", "backgroundImage": {"height": 464, "url": "web+graphie:${☣ LOCALPATH}/images/7ba401728d08a4c209b0423794bcbe7025215d78", "width": 464}, "box": [464, 464], "caption": "", "labels": [], "range": [[0, 10], [0, 10]], "static": false, "title": ""}, "static": false, "type": "image", "version": {"major": 0, "minor": 0}}, "matcher 1": {"alignment": "default", "graded": true, "options": {"labels": ["**Feature**", "**Statement**"], "left": ["$x$-intercept", "Relative maximum or minimum", "Increasing or decreasing interval"], "orderMatters": false, "padding": true, "right": ["When the width of the rectangle is $20\\text{ cm}$, it has no area.", "The greatest possible area of the rectangle is $100\\text{ sq. cm}$.", "As the width of the rectangle is extended towards $10\\text{ cm}$, the area of the rectangle grows."]}, "static": false, "type": "matcher", "version": {"major": 0, "minor": 0}}}}}