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You 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.

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This diagram is for illustration only and is not to scale

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Use Sine rule and Cosine rule to solve a Geometry problem in geology.

Using basic principles of trigonometry, find the height ($h$) of the mountain to the nearest metre.

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

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Find also the distance from your location to the centre of the mountain base ($d_{total}$) to the nearest metre.

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[[1]] $\\text{m}$

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Step one:

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To solve this problem you need to consider the triangle which can be constructed from the angles and movement described in the question statement:

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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 $\\var{thtchng}^\\circ$.

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It 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:

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• $C=180^\\circ-\\var{thetadeg}^\\circ-\\var{thtchng}^\\circ$
• \n
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Now, to find the value of $c$, we use the sine rule where:

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$\\big(\\frac{a}{\\sin(A)}\\big)=\\big(\\frac{b}{\\sin(B)}\\big)=\\big(\\frac{c}{\\sin(C)}\\big)$

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Rearranged with the known variables in terms of $c$ gives:

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$c=\\big(\\frac{a}{\\sin(A)}\\big)\\times\\sin(C)$

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$\\therefore,\\;\\;\\;\\; c=\\frac{\\var{deltf}}{\\sin(\\var{thtchng}^\\circ)}\\times \\sin((180^\\circ-\\var{thtchng}^\\circ-\\var{thetadeg}^\\circ))$ m

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$c$ =

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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 ($d_{total}$).

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From the options below, choose the two methods which can be adopted, considering the values you know and can substitute.

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(Note: Use the unrounded value for $c$ when making calculations for $h$ and $d_{total}$ to maintain accuracy.)

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SOHCAHTOA:

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$\\sin(\\theta)=\\big(\\frac{opp}{hyp}\\big),\\;\\;\\;\\;\\cos(\\theta)=\\big(\\frac{adj}{hyp}\\big),\\;\\;\\;\\;\\tan(\\theta)=\\big(\\frac{opp}{adj}\\big)$

", "

Pythagoras' theorem:

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$a^2+b^2=c^2$

", "

Sine rule method:

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$\\frac{a}{sin(A)}=\\frac{b}{sin(B)} =\\frac{c}{sin(C)}$

", "

Cosine rule method:

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$a^2=b^2+c^2-2bc\\;cos(A)$

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Hence, calculate the value of $h$ to the nearest metre:

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$h$ =

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Also, calculate $d_{total}$ to the nearest metre:

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$d_{total}$ =

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