// Numbas version: exam_results_page_options
{"name": "Applied y-intercepts: Investing in boats", "extensions": ["geogebra", "random_person"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "<p>This question provides an example of&nbsp;an initial bank account investment with a fixed return and tests the student's understanding of an application of intercepts.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["p", "friend", "m", "c", "bal"], "type": "question", "extensions": ["geogebra", "random_person"], "advice": "<h4>a)</h4>\n<p>The balance increases by &pound;{m} each year, so $m = \\var{m}$.</p>\n<p>We know that&nbsp;at $x = 9$ years, $y = \\var{bal}$. Substituting these values into the equation, we obtain</p>\n<p>\\[ \\var{bal} = \\var{m} \\times 9 + c \\]</p>\n<p>Rearrange this to find $c$:</p>\n<p>\\begin{align}<br/>c &amp;= \\var{bal} - \\var{m} \\times 9 \\\\ <br/>&amp;= \\var{bal}&nbsp;- \\var{m*9} \\\\<br/>&amp;= \\var{c}&nbsp;<br/>\\end{align}</p>\n<p>So the formula for the account balance is</p>\n<p>\\[y = \\var{m}x+\\var{c}\\text{.}\\]</p>\n<h4>b)</h4>\n<p>The constant term $\\var{c}$ determines the point at which the line crosses the $y$-axis. This point is&nbsp;called the <em>$y$-intercept</em>.</p>\n<h4>c)</h4>\n<p>The initial investment is the value of $y$ at $x = 0$, so it's &pound;{c}$.</p>\n<h4>d)</h4>\n<p>It is useful to plot the graph of {friend['name']}'s savings account against your own for comparison ({friend['name']}'s balance is shown as a dashed line):</p>\n<p>{geogebra_applet('gpFmg3Ex',[[\"p\",p],[\"m\",m],[\"c\",c]])}</p>\n<p>Using this graph,&nbsp;we can see that only two of the statements are true:</p>\n<ul>\n<li>The two lines are parallel. This is because they both increase by &pound;{m} each year.</li>\n<li>The plot of {friend['name']}'s balance has a higher $y$-intercept because {friend['pronouns']['their']} initial balance is higher.</li>\n</ul>\n<p>As the gradients of the two lines on the graph are the same, we can eliminate the other two statements about the lines converging and&nbsp;about having a higher gradient.</p>", "variable_groups": [], "rulesets": {}, "statement": "<p>You are a forgetful investor set on saving enough money to buy a new fishing boat for when you retire.</p>\n<p>Your savings account manager tells you your savings account is worth &pound;{formatnumber(bal,\"en\")}.&nbsp;</p>\n<p>You have forgotten&nbsp;the principal amount you started with the account with; however you do know that you have been saving for exactly nine years now and your manager informs you that the bank has been paying you a premium of &pound;{m} per year.</p>\n<p>Your account manager shows you this graph, which plots account balance over time for a given principal amount.</p>\n<p><em>The line on the graph below can be&nbsp;repositioned by dragging the slider.</em></p>\n<p>{geogebra_applet('HtnCWSSQ',[[\"p\",p],[\"m\",m],[\"c\",c]])}&nbsp;</p>", "name": "Applied y-intercepts: Investing in boats", "parts": [{"scripts": {}, "gaps": [{"showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "{m}x+{c}", "scripts": {}, "variableReplacements": [], "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}], "type": "gapfill", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "<p>With $y$ representing the account balance and $x$ the&nbsp;number of years the account has been open, give an&nbsp;expression for the balance in the form $y=mx+c$.</p>\n<p>$y=$&nbsp;[[0]]</p>\n<p><em></em></p>", "variableReplacementStrategy": "originalfirst"}, {"shuffleChoices": true, "type": "1_n_2", "minMarks": 0, "showCorrectAnswer": true, "prompt": "<p>Which of the following elements of the graph corresponds to the constant part of the equation?</p>", "showFeedbackIcon": true, "displayColumns": 0, "choices": ["<p>y-intercept</p>", "<p>x-intercept</p>", "<p>z-intercept</p>", "<p>The origin</p>"], "scripts": {}, "distractors": ["", "", "", ""], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "marks": 0, "variableReplacements": [], "matrix": ["1", 0, 0, 0], "displayType": "dropdownlist"}, {"scripts": {}, "gaps": [{"showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "{c}", "scripts": {}, "variableReplacements": [], "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}], "type": "gapfill", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "<p>From your answer to the last question, state the initial investment you made towards saving for your new fishing boat.</p>\n<p>Initial investment $=$ $&pound;$[[0]].</p>", "variableReplacementStrategy": "originalfirst"}, {"shuffleChoices": true, "type": "m_n_2", "maxAnswers": 0, "minMarks": 0, "showCorrectAnswer": true, "minAnswers": 0, "prompt": "<p>Your friend, {friend['name']}, is considerably wealthier than you are, so {friend['pronouns']['they']}&nbsp;{if(friend['gender']='neutral','start','starts')}&nbsp;with twice the investment you did but still {if(friend['gender']='neutral','receive','receives')} the same annual payment of $&pound;\\var{m}$.</p>\n<p>Which of the following statements comparing&nbsp;the graph of {friend['name']}'s&nbsp;account balance&nbsp;to yours are true?</p>", "showFeedbackIcon": true, "displayColumns": "1", "choices": ["<p>The gradient is equal and hence {friend['pronouns']['their']}&nbsp;line would be parallel to yours.</p>", "<p>The plot of {friend['pronouns']['their']} balance&nbsp;crosses the $y$-axis at a higher point than yours.</p>", "<p>The plot of {friend['pronouns']['their']} balance has a higher gradient.</p>", "<p>The plots of your balance and&nbsp;{friend['pronouns']['theirs']} cross at some point.</p>"], "scripts": {}, "distractors": ["", "", "", ""], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "marks": 0, "variableReplacements": [], "matrix": ["1", "1", 0, 0], "displayType": "checkbox", "warningType": "none"}], "tags": ["applications of y-intercepts", "taxonomy", "y-intercept"], "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"bal": {"description": "", "group": "Ungrouped variables", "definition": "m*9+c", "name": "bal", "templateType": "anything"}, "c": {"description": "", "group": "Ungrouped variables", "definition": "random(995,1005,1010,1015,1020,1025)", "name": "c", "templateType": "anything"}, "p": {"description": "<p>not intercept, starting intercept</p>", "group": "Ungrouped variables", "definition": "1100", "name": "p", "templateType": "anything"}, "friend": {"description": "", "group": "Ungrouped variables", "definition": "random_person()", "name": "friend", "templateType": "anything"}, "m": {"description": "", "group": "Ungrouped variables", "definition": "random(15,30,45)", "name": "m", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}]}