Question

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Explanation:

Identify the geometric setup

We analyze the given oblique triangle \(ABC\) representing the mountain and cable car system.

• Let \(D\) be the base of the mountain directly below the peak \(C\).
• Point \(A\) is the starting station, \(1.3\text{ miles}\) from the base \(D\). Thus, \(AD = 1.3\text{ miles}\).
• Converting miles to feet:

$$ AD = 1.3 \times 5280 = 6864\text{ feet} $$

• The angle of elevation from \(A\) to the peak \(C\) is \(20^\circ\), so \(\angle CAD = 20^\circ\).
• The angle of elevation of the mountain's slope from the base \(B\) (or a point along the ground) to the peak \(C\) is \(64^\circ\). Since \(B\) lies between \(A\) and \(D\), the angle of elevation at \(B\) is \(\angle CBD = 64^\circ\).
• Therefore, in the oblique triangle \(ABC\), the interior angle at \(B\) is:

$$ \angle ABC = 180^\circ - 64^\circ = 116^\circ $$

Determine the third angle

Using the Triangle Angle Sum theorem on oblique triangle \(ABC\):

• The sum of angles in \(\triangle ABC\) is \(180^\circ\).
• We calculate the angle at the peak, \(\angle ACB\):

$$ \angle ACB = 180^\circ - 20^\circ - 116^\circ = 44^\circ $$

Calculate the cable car distance

Using the Law of Sines on oblique triangle \(ABC\) to find the cable car distance \(AC\) (labeled as \(b\)):

• The side opposite \(\angle ABC\) is \(AC = b\).
• The side opposite \(\angle ACB\) is \(AB = c\).
• First, we find the ground distance \(AB\). Since \(AD = 6864\text{ feet}\) and \(BD = \frac{CD}{\tan(64^\circ)}\), we can set up the system.
• Alternatively, using right triangles:
• In right triangle \(ADC\): \(CD = AD \tan(20^\circ) = 6864 \tan(20^\circ)\)
• Let's compute the height \(CD\):

$$ CD = 6864 \times 0.36397 = 2498.3\text{ feet} $$

• In right triangle \(BDC\): \(BD = \frac{CD}{\tan(64^\circ)} = \frac{2498.3}{2.0503} = 1218.5\text{ feet}\)
• Thus, the ground distance \(AB\) is:

$$ AB = AD - BD = 6864 - 1218.5 = 5645.5\text{ feet} $$

• Now, apply the Law of Sines to find \(AC\):

$$ \frac{AC}{\sin(116^\circ)} = \frac{AB}{\sin(44^\circ)} \implies AC = 5645.5 \times \frac{\sin(116^\circ)}{\sin(44^\circ)} $$

$$ AC = 5645.5 \times \frac{0.89879}{0.69466} \approx 7304.2\text{ feet} $$

Calculate the distance from A to B

We already computed the ground distance \(AB\) in the previous step:

• \(AB = 5645.5\text{ feet}\).
• To verify using the Law of Sines directly:

$$ \frac{BC}{\sin(20^\circ)} = \frac{AB}{\sin(44^\circ)} \implies BC = 5645.5 \times \frac{\sin(20^\circ)}{\sin(44^\circ)} \approx 2779.6\text{ feet} $$

Find the height of the mountain

Using the right triangle \(ADC\) or \(BDC\):

• Height \(h = CD = AD \tan(20^\circ)\)
• Using \(AD = 6864\text{ feet}\):

$$ h = 6864 \times \tan(20^\circ) \approx 2498.3\text{ feet} $$

Answer:

a. The distance covered by the cable car is 7304.2 feet.

b. The distance \(AB\) in oblique triangle \(ABC\) is 5645.5 feet.

c. The height of the mountain is 2498.3 feet.