To find the length of the cable securing the telephone tower, we can model the situation using a right triangle. Here's how you can set up the problem:
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Identify the Triangle: The telephone tower, the shadow, and the cable form a right triangle. The height of the tower represents the opposite side, the length of the shadow represents the adjacent side, and the cable represents the hypotenuse of the triangle.
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Label the Sides:
- Let \( h \) be the height of the tower (the opposite side to the angle).
- The length of the shadow is 50 feet (the adjacent side).
- The angle between the ground and the cable is 48 degrees (the angle of elevation).
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Use Trigonometric Relationships: You can use the cosine function to find the length of the cable (hypotenuse). The cosine function relates the adjacent and hypotenuse:
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
where \( \theta \) is the angle (48 degrees) and \( \text{Adjacent} \) is the length of the shadow (50 feet).
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Set Up the Equation: Let \( c \) be the length of the cable (hypotenuse). Then,
\[ \cos(48^\circ) = \frac{50}{c} \]
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Solving for \( c \): Rearranging the equation to solve for \( c \):
\[ c = \frac{50}{\cos(48^\circ)} \]
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Calculate: Now you can use a calculator to find \( \cos(48^\circ) \) and then calculate \( c \):
- Calculate \( \cos(48^\circ) \).
- Then compute \( c \).
This will yield the length of the cable that secures the telephone tower to the ground. Be sure to see if your calculator is in the correct mode (degree mode) before calculating.
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