// Numbas version: exam_results_page_options {"name": "4: Trigonometry: MOUNTAIN PART 2 (follows on from part 1)", "extensions": [], "custom_part_types": [], "resources": [["question-resources/2016-08-11.png", "/srv/numbas/media/question-resources/2016-08-11.png"], ["question-resources/2016-08-11_8.png", "/srv/numbas/media/question-resources/2016-08-11_8.png"], ["question-resources/2016-08-11_13.png", "/srv/numbas/media/question-resources/2016-08-11_13.png"], ["question-resources/2016-08-16_5.png", "/srv/numbas/media/question-resources/2016-08-16_5.png"], ["question-resources/2016-08-17_8.png", "/srv/numbas/media/question-resources/2016-08-17_8.png"], ["question-resources/2016-08-17_9.png", "/srv/numbas/media/question-resources/2016-08-17_9.png"], ["question-resources/newheli.png", "/srv/numbas/media/question-resources/newheli.png"], ["question-resources/newheli2.png", "/srv/numbas/media/question-resources/newheli2.png"], ["question-resources/newheli3.png", "/srv/numbas/media/question-resources/newheli3.png"], ["question-resources/basicscenario.png", "/srv/numbas/media/question-resources/basicscenario.png"], ["question-resources/newheli4.png", "/srv/numbas/media/question-resources/newheli4.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"description": "

set up x- and y axises.

set linear equations and solve the simulatenous equations.

", "licence": "Creative Commons Attribution 4.0 International"}, "name": "4: Trigonometry: MOUNTAIN PART 2 (follows on from part 1)", "variable_groups": [{"variables": ["bcdy", "bcdd_2dp", "bcd", "bca", "bcdd", "h2dp", "dtotal2dp", "grad1", "grad2"], "name": "line eq"}, {"variables": ["coordx", "simp", "coordy"], "name": "sim eq"}, {"variables": ["vel", "hypot"], "name": "velocity"}, {"variables": ["h2", "b2", "r2", "thetam", "thetaf", "thetafdeg", "f", "dtot", "xspot", "yspot", "cspot", "vt", "m1", "m2", "p", "thetafrnd", "cm", "f2", "flare"], "name": "Linked variables"}], "functions": {}, "statement": "

After observing the mountain, you walk back to base. You fall over at base and need medical assistance immediately. You call for help and a rescue helicopter comes to find you. The pilot cannot see you initially due to the mountain blocking their view.

This problem challenges you to use your knowledge of trigonometry and linear equations to find how long it takes before the helicopter pilot is able to see you from take-off. The parts and the steps below will guide you through the process. Alternatively, you could attempt the problem without reading the steps.

The facts provided are:

The helicopter sets off at angle $\\theta$ of $\\var{thetafrnd}^\\circ$ from the ground, which remains constant during its ascent.
The helicopter travels along this flight path at an average speed, $v$, of $\\var{vel}\\text{ms}^{-1}$.
The height of the mountain, $h$, is $\\var{h2}\\text{m}$.
The distance from the rescue base to the ground-level centre of the mountain, $r$, is $\\var{r2}\\text{m}$.
The distance between base (where you are) and the ground-level centre of the mountain, $d_{total}$_, is $\\var{dtot}\\text{m}$.

Note: Due to the multi-step process to finding a solution, it is advisable that you find the algebraic expressions for all values which are to be determined along the way so that given values can be input into equations at the end, thus decreasing the likelihood of rounding errors.

", "parts": [{"prompt": "

Use the ground as x-axis, and the line for the height of mountain as y-axis. Find the line equations involved. Keep the accuracy to two decimal places when the number isn't an integer:

Line equation of the path of vision:

$y=$ [[0]] $x$+ [[2]]

Line equation of the helicopter flight path:

$y=$ [[1]] $x$+ [[3]]

", "variableReplacements": [], "stepsPenalty": 0, "type": "gapfill", "gaps": [{"correctAnswerStyle": "plain", "precisionPartialCredit": 0, "mustBeReduced": false, "maxValue": "m1+0.04", "strictPrecision": true, "precisionType": "dp", "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "precision": "2", "showFeedbackIcon": true, "showCorrectAnswer": true, "precisionMessage": "

You have not given your answer to the correct precision.

", "mustBeReducedPC": 0, "marks": 1, "type": "numberentry", "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "m1-0.04", "variableReplacements": [], "correctAnswerFraction": false, "showPrecisionHint": false}, {"correctAnswerStyle": "plain", "precisionPartialCredit": "100", "mustBeReduced": false, "maxValue": "m2+0.04", "strictPrecision": true, "precisionType": "dp", "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "precision": "2", "showFeedbackIcon": true, "showCorrectAnswer": true, "precisionMessage": "

You have not given your answer to the correct precision.

", "mustBeReducedPC": 0, "marks": 1, "type": "numberentry", "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "m2-0.04", "variableReplacements": [], "correctAnswerFraction": false, "showPrecisionHint": false}, {"correctAnswerStyle": "plain", "precisionPartialCredit": "100", "mustBeReduced": false, "maxValue": "h2+0.04", "strictPrecision": false, "precisionType": "dp", "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "precision": "2", "showFeedbackIcon": true, "showCorrectAnswer": true, "precisionMessage": "

You have not given your answer to the correct precision.

", "mustBeReducedPC": 0, "marks": 1, "type": "numberentry", "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "h2-0.04", "variableReplacements": [], "correctAnswerFraction": false, "showPrecisionHint": false}, {"correctAnswerStyle": "plain", "precisionPartialCredit": 0, "mustBeReduced": false, "maxValue": "p+0.04", "strictPrecision": true, "precisionType": "dp", "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "precision": "2", "showFeedbackIcon": true, "showCorrectAnswer": true, "precisionMessage": "

You have not given your answer to the correct precision.

", "mustBeReducedPC": 0, "marks": 1, "type": "numberentry", "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "p-0.04", "variableReplacements": [], "correctAnswerFraction": false, "showPrecisionHint": false}], "scripts": {}, "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Step one:

The most important beginning stage is to set a reference frame. Thus, the x-axis is defined as ground-level and the y-axis is the line which moves vertically through the ground-level centre of the mountain and its peak.

Step two:

To work out the equation for the path of vision you need to find the gradient $m$ and the y-intercept $c$ to form the known linear equation structure of $y=mx+c$.

To do this, form a right-angled triangle between your position, the ground-level centre of the mountain and the mountain peak. The gradient, or $\\frac{rise}{run}$, is taken from $\\frac{h}{d_{total}}$. The change in $x$ is negative and thus the gradient also takes a negative value. The y-intercept is the point where the path of vision crosses the y-axis. This is at the peak and thus takes the value of the height of the mountain. The equation is found to be:

$y=-\\big(\\frac{h}{d_{total}}\\big)x+h$

Step three:

You now need to work out the line equation of the helicopter's flight path.

Let the height of the helicopter when directly above the peak, or in other words the y-coordinate at which the flight path crosses the y-axis, be $p$.

$p$ can be calculated by taking the fact that $\\tan(\\theta)=\\frac{p}{r}\\;\\;\\therefore\\;\\;p=r\\;\\tan(\\theta)$

This gives both the gradient and y-intercept of the line to provide the equation:

$y=\\tan(\\theta)x+r\\tan(\\theta)$

Now that you have two line equations for the different paths, you can solve simultaneously to find the point at which you are visible to the helicopter.

The coordinates of this point are ([[0]], [[1]]), giving the values correct to two decimal places.

Note: Use the algebraic expressions for $m$ and $c$ in the line equations when developing an expression for these coordinates. This will avoid unecessary rounding errors in the final answer.

", "variableReplacements": [], "stepsPenalty": 0, "type": "gapfill", "gaps": [{"correctAnswerStyle": "plain", "precisionPartialCredit": 0, "mustBeReduced": false, "maxValue": "xspot+0.1", "strictPrecision": false, "precisionType": "dp", "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "precision": "2", "showFeedbackIcon": true, "showCorrectAnswer": true, "precisionMessage": "

You have not given your answer to the correct precision.

", "mustBeReducedPC": 0, "marks": 1, "type": "numberentry", "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "xspot-0.1", "variableReplacements": [], "correctAnswerFraction": false, "showPrecisionHint": false}, {"correctAnswerStyle": "plain", "precisionPartialCredit": 0, "mustBeReduced": false, "maxValue": "yspot+0.1", "strictPrecision": true, "precisionType": "dp", "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "precision": "2", "showFeedbackIcon": true, "showCorrectAnswer": true, "precisionMessage": "

You have not given your answer to the correct precision.

", "mustBeReducedPC": 0, "marks": 1, "type": "numberentry", "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "yspot-0.1", "variableReplacements": [], "correctAnswerFraction": false, "showPrecisionHint": false}], "scripts": {}, "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

You want to find the coordinates that give the point $I$ where the two lines cross, as this is the point at which the helicopter can see you. To find this point, you need to construct a simultaneous equation involving the two line equations to solve for $x$ and $y$. The most appropriate method: substitution.

At the point of intersection (let $d_{total}=d$):

$y=\\tan(\\theta)x+r\\tan(\\theta)$ and $y=-\\big(\\frac{h}{d}\\big)x+h$

$\\therefore\\;\\;,\\;\\;\\tan(\\theta)x+r\\tan(\\theta)=-\\big(\\frac{h}{d}\\big)x+h$

This is then rearranged in terms of $x$,

$x\\big(\\tan(\\theta)+\\frac{h}{d}\\big)=h-r\\tan(\\theta)$

$\\therefore\\;\\;,\\;\\;x=\\Big(\\frac{h-r\\tan(\\theta)}{\\tan(\\theta)+\\frac{h}{d}}\\Big)$

The algebraic value for $x$ can then be substituted into one of the two line equations, and simplified in terms of $y$.

$y=-\\big(\\frac{h}{d}\\big)\\Big(\\frac{h-r\\tan(\\theta)}{\\tan(\\theta)+\\frac{h}{d}}\\Big)+h$

$y=h\\Big(\\frac{r\\tan(\\theta)-h}{d\\tan(\\theta)+h}+1\\Big)=h\\Big(\\frac{r\\tan(\\theta)+d\\tan(\\theta)}{d\\tan(\\theta)+h}\\Big)$

$\\therefore\\;\\;,\\;\\;y=h\\tan(\\theta)\\Big(\\frac{r+d}{d\\tan(\\theta)+h}\\Big)$

When $x$ and $y$ are evaluated in this way, the coordinates of point $I$ are found.

Finally, find how long it takes for the helicopter to be able to see you from setting off at the rescue base. Give your answer correct to the nearest second.

[[0]] $\\text{seconds}$

", "variableReplacements": [], "stepsPenalty": 0, "type": "gapfill", "gaps": [{"mustBeReduced": false, "maxValue": "vt+1", "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "marks": 1, "type": "numberentry", "allowFractions": true, "variableReplacementStrategy": "originalfirst", "minValue": "vt-1", "variableReplacements": [], "correctAnswerFraction": false}], "scripts": {}, "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

You must first find the distance, $f$, that the helicopter travels from the rescue base until the intersection of the two paths at point $I$.

This can be found from the fact that:

$f\\sin(\\theta)=j\\;\\;\\therefore\\;\\;f=\\big(\\frac{j}{\\sin(\\theta)}\\big)$

We can also take the value of $j$ as the y-coordinate of the intersection point $I$.

Thus, $f$ is found to be:

$f=\\Big(\\frac{h\\tan(\\theta)\\big(\\frac{r+d}{d\\tan(\\theta)+h}\\big)}{\\sin(\\theta)}\\Big)=\\Big(\\frac{h\\tan(\\theta)(r+d)}{\\sin(\\theta)(d\\tan(\\theta)+h)}\\Big)$

This can be further simplified to leave:

$f=\\left(\\frac{h(r+d)}{d\\sin(\\theta)+h\\cos(\\theta)}\\right)$

We then take the equation relating distance, speed and time and rearrange in terms of time. In this case, the distance is $f$ and the speed $v$ is that which is given in the original question statement of $\\var{vel}\\text{ms}^{-1}$.

$v=\\frac{f}{t}\\;\\;,\\;\\;t=\\frac{f}{v}$

Thus the time taken for the helicopter pilot to see you from taking-off is given by:

$t=\\left(\\frac{h(r+d)}{v\\;(d\\sin(\\theta)+h\\cos(\\theta))}\\right)$

", "variableReplacements": [], "type": "information", "scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0, "showFeedbackIcon": true}], "showCorrectAnswer": true, "marks": 0, "showFeedbackIcon": true}], "ungrouped_variables": ["x", "a", "h", "dtotal", "b", "y"], "preamble": {"css": "", "js": ""}, "variables": {"grad1": {"description": "

gradient of line one - persons vision

", "templateType": "anything", "name": "grad1", "definition": "precround(-{h2dp}/{dtotal2dp},2)", "group": "line eq"}, "r2": {"description": "", "templateType": "anything", "name": "r2", "definition": "(b2/2)*random(2.1..2.5#0.05)", "group": "Linked variables"}, "flare": {"description": "", "templateType": "anything", "name": "flare", "definition": "precround(vt*random(0.8..1.2#0.1),0)", "group": "Linked variables"}, "h2": {"description": "", "templateType": "anything", "name": "h2", "definition": "random(1000..3000)", "group": "Linked variables"}, "grad2": {"description": "

gradient of line two - helicpters path

", "templateType": "anything", "name": "grad2", "definition": "precround({bcdy}/{bcdd_2dp},2)", "group": "line eq"}, "xspot": {"description": "", "templateType": "anything", "name": "xspot", "definition": "precround(((h2-r2*tan(thetaf))/(tan(thetaf)+(h2/dtot))),2)", "group": "Linked variables"}, "thetafdeg": {"description": "", "templateType": "anything", "name": "thetafdeg", "definition": "degrees(thetaf)", "group": "Linked variables"}, "bcdy": {"description": "

height at which the helicopter crosses the y-axis

", "templateType": "anything", "name": "bcdy", "definition": "precround(tan({bca}/180*pi)*{bcdd_2dp},2)", "group": "line eq"}, "b2": {"description": "", "templateType": "anything", "name": "b2", "definition": "h2*random(0.4..0.8#0.1)", "group": "Linked variables"}, "vt": {"description": "", "templateType": "anything", "name": "vt", "definition": "precround(cspot/vel,0)", "group": "Linked variables"}, "y": {"description": "

resultant angle to go with b and a (helps work out the overall height)

", "templateType": "anything", "name": "y", "definition": "(180-{x}-5)", "group": "Ungrouped variables"}, "bcdd_2dp": {"description": "

distance from rescue base to the centre of the mountain at ground level to 2dp

", "templateType": "anything", "name": "bcdd_2dp", "definition": "precround({bcdd},2)", "group": "line eq"}, "simp": {"description": "", "templateType": "anything", "name": "simp", "definition": "(precround(-(h2dp/dtotal2dp),2))-(precround(bcdy/bcdd_2dp,2))", "group": "sim eq"}, "f2": {"description": "", "templateType": "anything", "name": "f2", "definition": "(h2*(r2+dtot))/(dtot*sin(thetaf)+h2*cos(thetaf))", "group": "Linked variables"}, "dtot": {"description": "", "templateType": "anything", "name": "dtot", "definition": "(b2/2)*random(1.5..2.5#0.1 except r2*2/b2 except 2)", "group": "Linked variables"}, "thetaf": {"description": "", "templateType": "anything", "name": "thetaf", "definition": "radians(thetam+random(1..3#0.5))", "group": "Linked variables"}, "a": {"description": "

need as intermediate to work out height

", "templateType": "anything", "name": "a", "definition": "(20/(sin(5/180*pi)))*sin({x}/180*pi)", "group": "Ungrouped variables"}, "bcd": {"description": "

distance between the rescue camp and the top of the mountain

", "templateType": "anything", "name": "bcd", "definition": "random(250..300)", "group": "line eq"}, "x": {"description": "

angle subtended by the mountain and the ground

", "templateType": "randrange", "name": "x", "definition": "random(40..50#1)", "group": "Ungrouped variables"}, "cspot": {"description": "", "templateType": "anything", "name": "cspot", "definition": "h2*sec(thetaf)*((r2+dtot)/(dtot*tan(thetaf)+h2))", "group": "Linked variables"}, "f": {"description": "", "templateType": "anything", "name": "f", "definition": "(r2/cos(thetaf))", "group": "Linked variables"}, "bcdd": {"description": "

distance from rescue base to the centre of the mountain at ground level

", "templateType": "anything", "name": "bcdd", "definition": "sqrt({bcd}^2-{h2dp}^2)", "group": "line eq"}, "m1": {"description": "", "templateType": "anything", "name": "m1", "definition": "precround(-(h2/dtot),2)", "group": "Linked variables"}, "coordy": {"description": "

coordinate y of intersection

", "templateType": "anything", "name": "coordy", "definition": "((bcdy/bcdd_2dp)*coordx)+bcdy", "group": "sim eq"}, "b": {"description": "

need as intermediate to work out heihgt

", "templateType": "anything", "name": "b", "definition": "(a/sin({x}/180*pi))*sin({y}/180*pi)", "group": "Ungrouped variables"}, "coordx": {"description": "

coordinate x of intersectin

", "templateType": "anything", "name": "coordx", "definition": "(bcdy-h2dp)/simp", "group": "sim eq"}, "thetam": {"description": "", "templateType": "anything", "name": "thetam", "definition": "degrees(arctan(h2/r2))", "group": "Linked variables"}, "hypot": {"description": "", "templateType": "anything", "name": "hypot", "definition": "sqrt(({bcdd_2dp}+{coordx}^2)+coordy^2)", "group": "velocity"}, "h": {"description": "

heihgt of mountain

", "templateType": "anything", "name": "h", "definition": "(b/sin(90/180*pi))*sin({x}/180*pi)", "group": "Ungrouped variables"}, "dtotal2dp": {"description": "

distance between centre of mountain at ground level and person to 2dp

", "templateType": "anything", "name": "dtotal2dp", "definition": "precround(dtotal,2)", "group": "line eq"}, "yspot": {"description": "", "templateType": "anything", "name": "yspot", "definition": "precround(h2*tan(thetaf)*((r2+dtot)/(dtot*tan(thetaf)+h2)),2)", "group": "Linked variables"}, "thetafrnd": {"description": "", "templateType": "anything", "name": "thetafrnd", "definition": "precround(thetafdeg,4)", "group": "Linked variables"}, "p": {"description": "", "templateType": "anything", "name": "p", "definition": "precround(r2*tan(thetaf),2)", "group": "Linked variables"}, "bca": {"description": "

angle the helicopter sets off at

", "templateType": "randrange", "name": "bca", "definition": "random(50..60#1)", "group": "line eq"}, "dtotal": {"description": "

distance centre of mountain and person

", "templateType": "anything", "name": "dtotal", "definition": "sqrt(b^2-h^2)", "group": "Ungrouped variables"}, "cm": {"description": "", "templateType": "anything", "name": "cm", "definition": "precround(r2/cos(arctan(h2/r2)),4)", "group": "Linked variables"}, "m2": {"description": "", "templateType": "anything", "name": "m2", "definition": "precround(tan(thetaf),2)", "group": "Linked variables"}, "vel": {"description": "

velocity of helicopter

", "templateType": "anything", "name": "vel", "definition": "random(5..10)", "group": "velocity"}, "h2dp": {"description": "

height of mountain 2dp

", "templateType": "anything", "name": "h2dp", "definition": "precround(h,2)", "group": "line eq"}}, "extensions": [], "rulesets": {}, "advice": "

Once completed, click 'Try another question like this one' to reset the variables provided in the question statement. There are ways of reaching the final answers other than those outlined in the steps. Experiment and see if you can find another method.

", "tags": [], "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/"}]}