![](\"resources/question-resources/2016-08-11_8.png\")
This diagram is for illustration only and is not to scaleYou are observing the lithology of a small snowy mountain nearby. You are stood at 'base' which is $\\var{f}\\text{m}$ away ($d$) from the foot of the mountain. The angle subtended by the top of the mountain with the ground where you are stood at base is $\\var{thetadeg}^\\circ$ ($x$). You walk $\\var{deltf}\\text{m}$ closer and angle $x$ becomes $\\var{thtchng}^\\circ$ larger.
\nThis diagram is for illustration only and is not to scale
Use Sine rule and Cosine rule to solve a Geometry problem in geology.
"}, "advice": "Use the steps for advice.
", "name": "4: Trigonometry - MOUNTAIN PART 1 (see part two for follow on)", "variablesTest": {"maxRuns": 100, "condition": ""}, "ungrouped_variables": ["x", "x_5", "d", "d_20", "y", "z", "a", "b", "c", "h", "dtotal"], "functions": {}, "variable_groups": [{"name": "Unnamed group", "variables": ["bcdy", "bcdd_2dp", "bcd", "bca", "bcdd"]}, {"name": "Linked variables", "variables": ["thetarad", "thetadeg", "f", "m", "h2", "alph", "bet", "deltf", "theta2", "thetchnge", "thtchng", "hfinal", "dfinal", "c2", "h3", "alph2", "thtchrad"]}], "rulesets": {}, "parts": [{"stepsPenalty": 0, "marks": 0, "gaps": [{"allowFractions": false, "maxValue": "hfinal", "minValue": "hfinal", "showFeedbackIcon": true, "variableReplacements": [], "correctAnswerFraction": false, "scripts": {}, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "type": "numberentry"}, {"allowFractions": false, "maxValue": "dfinal", "minValue": "dfinal", "showFeedbackIcon": true, "variableReplacements": [], "correctAnswerFraction": false, "scripts": {}, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "type": "numberentry"}], "variableReplacements": [], "prompt": "Using basic principles of trigonometry, find the height ($h$) of the mountain to the nearest metre.
\n[[0]] $\\text{m}$
\nFind also the distance from your location to the centre of the mountain base ($d_{total}$) to the nearest metre.
\n[[1]] $\\text{m}$
", "showCorrectAnswer": true, "showFeedbackIcon": true, "steps": [{"allowFractions": false, "maxValue": "c2", "minValue": "c2", "showFeedbackIcon": true, "variableReplacements": [], "prompt": "Step one:
\nTo solve this problem you need to consider the triangle which can be constructed from the angles and movement described in the question statement:
\n![](\"resources/question-resources/sineruletri.png\")
Since alternate angles (Z angles) are equal, the bottom right hand angle $B$ is equal to $x$ ($\\var{thetadeg}^\\circ$), as provided in the question statement. The side $a$ is the distance moved towards the mountain and angle $A$ is the increase in angle subtended from the mountain peak to the new position, which in this case is $\\var{thtchng}^\\circ$.
\nIt may be useful to also consider that angles in a triangle add up to $180^\\circ$, therefore allowing the value of the bottom left hand angle to be found as follows:
\nNow, to find the value of $c$, we use the sine rule where:
\n$\\big(\\frac{a}{\\sin(A)}\\big)=\\big(\\frac{b}{\\sin(B)}\\big)=\\big(\\frac{c}{\\sin(C)}\\big)$
\nRearranged with the known variables in terms of $c$ gives:
\n$c=\\big(\\frac{a}{\\sin(A)}\\big)\\times\\sin(C)$
\n$\\therefore,\\;\\;\\;\\; c=\\frac{\\var{deltf}}{\\sin(\\var{thtchng}^\\circ)}\\times \\sin((180^\\circ-\\var{thtchng}^\\circ-\\var{thetadeg}^\\circ))$ m
\nGiving your answer to the nearest metre:
\n$c$ =
", "correctAnswerFraction": false, "scripts": {}, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "type": "numberentry"}, {"variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "variableReplacements": [], "displayColumns": 0, "prompt": "Once you have the value of the hypotenuse $c$, you can construct a triangle like the one below to determine the height of the mountain, $h$ and the distance from the starting position to the centre of the mountain base ($d_{total}$).
\n![](\"resources/question-resources/2016-08-16_4.png\"/)
From the options below, choose the two methods which can be adopted, considering the values you know and can substitute.
\n(Note: Use the unrounded value for $c$ when making calculations for $h$ and $d_{total}$ to maintain accuracy.)
", "minMarks": 0, "minAnswers": 0, "marks": 0, "scripts": {}, "warningType": "none", "shuffleChoices": false, "choices": ["SOHCAHTOA:
\n$\\sin(\\theta)=\\big(\\frac{opp}{hyp}\\big),\\;\\;\\;\\;\\cos(\\theta)=\\big(\\frac{adj}{hyp}\\big),\\;\\;\\;\\;\\tan(\\theta)=\\big(\\frac{opp}{adj}\\big)$
", "Pythagoras' theorem:
\n$a^2+b^2=c^2$
", "Sine rule method:
\n$\\frac{a}{sin(A)}=\\frac{b}{sin(B)} =\\frac{c}{sin(C)}$
", "Cosine rule method:
\n$a^2=b^2+c^2-2bc\\;cos(A)$
"], "maxAnswers": "2", "matrix": ["1", 0, "1", 0], "showCorrectAnswer": true, "maxMarks": "2", "distractors": ["", "", "", ""], "displayType": "checkbox", "type": "m_n_2"}, {"allowFractions": false, "maxValue": "hfinal", "minValue": "hfinal", "showFeedbackIcon": true, "variableReplacements": [], "prompt": "Hence, calculate the value of $h$ to the nearest metre:
\n$h$ =
", "correctAnswerFraction": false, "scripts": {}, "marks": "1", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "type": "numberentry"}, {"allowFractions": false, "maxValue": "dfinal", "minValue": "dfinal", "showFeedbackIcon": true, "variableReplacements": [], "prompt": "Also, calculate $d_{total}$ to the nearest metre:
\n$d_{total}$ =
", "correctAnswerFraction": false, "scripts": {}, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "type": "numberentry"}], "scripts": {}, "type": "gapfill", "variableReplacementStrategy": "originalfirst"}], "preamble": {"js": "", "css": ""}, "type": "question", "contributors": [{"name": "Sarah Turner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/881/"}]}]}], "contributors": [{"name": "Sarah Turner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/881/"}]}