Two cables, with lengths of 40 feet and 35 feet, are tied from the top of a pole to stakes in
the ground. If the stakes are 30 feet apart, find the height of the pole.

Answer:

The height of the pole is 33.89 feet.

Step-by-step explanation:

Since the length of the cables are 35 feet and 40 feet respectively, the 40 feet cable being farthest from the pole. We know that they are 30 feet apart. This forms a scalene triangle in which the angle between the 40 feet cable and the distance apart of the cables ( or ground) is the angle of elevation of the pole from the 40 feet cable.

So, we use the cosine rule to find this angle of elevation. So,

b² = a² + c² - 2accosB where a = 40 feet , b = 35 feet, c = 30 feet and B = angle of elevation of top of pole from 40 feet cable.

cosB = (a² + c² - b²)/2ac

cosB = (40² + 30² - 35²)/(2 × 40 × 30)

cosB = (1600 + 900 - 1225)/2400

cosB = 1275/2400

cosB = 0.53125

B = cos⁻¹(0.53125)

B = 57.9°

Since we now have the angle of elevation of the top of the pole from the 40 feet cable, we now find the height h of the pole from

sinB = h/40 since the 40 feet cable, the ground and the pole form a right-angled triangle.

h = 40sinB

h = 40sin57.9°

h = 40 × 0.8472

h = 33.89 feet

So, the height of the pole is 33.89 feet.

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