To find the length of the cable that secures the tower to the ground, we need to use trigonometry. The scenario you described involves a right triangle created by the height of the tower, the length of the shadow, and the cable acting as the hypotenuse.
Let's denote:
- \( h \) as the height of the tower,
- \( s \) as the length of the shadow (which is 50 feet),
- \( c \) as the length of the cable.
Given that the angle between the ground and the line from the top of the tower to the end of the shadow is \( 48^\circ \), we can set up the following relationships:
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To find the height of the tower \( h \): \[ \tan(48^\circ) = \frac{h}{s} \] Rearranging for \( h \) gives us: \[ h = s \cdot \tan(48^\circ) = 50 \cdot \tan(48^\circ) \]
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To find the length of the cable \( c \) (the hypotenuse of the triangle): We can use the cosine of the angle: \[ \cos(48^\circ) = \frac{s}{c} \] Rearranging for \( c \): \[ c = \frac{s}{\cos(48^\circ)} = \frac{50}{\cos(48^\circ)} \]
If Fiona wants to find the length of the cable \( c \), she should use the formula: \[ c = \frac{50}{\cos(48^\circ)} \]
Therefore, the correct setup among the options you provided is not explicitly listed; however, she should calculate \( c \) using the cosine function as shown above.
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