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Sagot :
To find the height of the telephone pole, we shall use trigonometric principles. When given an angle of elevation and the horizontal distance from the observer to the base of the object, we can use the tangent function, which relates an angle in a right triangle to the lengths of the opposite side (height of the pole) and the adjacent side (distance from the pole).
Here are the steps to solve the problem:
1. Identify the known values:
- Distance from the observer to the pole (adjacent side): 36 feet
- Angle of elevation: [tex]\(30^{\circ}\)[/tex]
2. Use the tangent function:
The tangent of an angle in a right triangle is the ratio of the opposite side (height of the pole) to the adjacent side (distance from the pole). This can be expressed as:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Plugging in the values we have:
[tex]\[ \tan(30^{\circ}) = \frac{\text{height of the pole}}{36} \][/tex]
3. Find the tangent of [tex]\(30^{\circ}\)[/tex]:
From trigonometric tables or calculator, we know:
[tex]\[ \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \][/tex]
4. Solve for the height of the pole:
[tex]\[ \frac{1}{\sqrt{3}} = \frac{\text{height of the pole}}{36} \][/tex]
To isolate the height of the pole, multiply both sides by 36:
[tex]\[ \text{height of the pole} = 36 \times \frac{1}{\sqrt{3}} \][/tex]
Simplifying further:
[tex]\[ \text{height of the pole} = 36 \div \sqrt{3} \][/tex]
5. Rationalize the denominator:
Multiply the numerator and denominator by [tex]\(\sqrt{3}\)[/tex]:
[tex]\[ \text{height of the pole} = \frac{36 \sqrt{3}}{3} = 12 \sqrt{3} \][/tex]
Thus, the height of the pole is [tex]\(12 \sqrt{3}\)[/tex] feet.
When comparing the height we calculated to the provided choices, we see that the correct answer matches one of the choices given:
[tex]\[12 \sqrt{3} \text{ ft}\][/tex]
Here are the steps to solve the problem:
1. Identify the known values:
- Distance from the observer to the pole (adjacent side): 36 feet
- Angle of elevation: [tex]\(30^{\circ}\)[/tex]
2. Use the tangent function:
The tangent of an angle in a right triangle is the ratio of the opposite side (height of the pole) to the adjacent side (distance from the pole). This can be expressed as:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Plugging in the values we have:
[tex]\[ \tan(30^{\circ}) = \frac{\text{height of the pole}}{36} \][/tex]
3. Find the tangent of [tex]\(30^{\circ}\)[/tex]:
From trigonometric tables or calculator, we know:
[tex]\[ \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \][/tex]
4. Solve for the height of the pole:
[tex]\[ \frac{1}{\sqrt{3}} = \frac{\text{height of the pole}}{36} \][/tex]
To isolate the height of the pole, multiply both sides by 36:
[tex]\[ \text{height of the pole} = 36 \times \frac{1}{\sqrt{3}} \][/tex]
Simplifying further:
[tex]\[ \text{height of the pole} = 36 \div \sqrt{3} \][/tex]
5. Rationalize the denominator:
Multiply the numerator and denominator by [tex]\(\sqrt{3}\)[/tex]:
[tex]\[ \text{height of the pole} = \frac{36 \sqrt{3}}{3} = 12 \sqrt{3} \][/tex]
Thus, the height of the pole is [tex]\(12 \sqrt{3}\)[/tex] feet.
When comparing the height we calculated to the provided choices, we see that the correct answer matches one of the choices given:
[tex]\[12 \sqrt{3} \text{ ft}\][/tex]
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