To solve the problem, we will use the properties of right triangles.
First, let's consider the two ropes and the angles they make with the ground.
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Rope 1: Forms a 45° angle.
- The height of the balloon (15 feet) can be related to the horizontal distance \(d_1\) from the base of the balloon to the ground directly below it using the tangent function. In a right triangle: \[ \tan(45^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{15}{d_1} \] Since \(\tan(45^\circ) = 1\), we get: \[ 1 = \frac{15}{d_1} \implies d_1 = 15 \text{ feet} \]
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Rope 2: Forms a 30° angle.
- We can relate the height of the balloon to the horizontal distance \(d_2\): \[ \tan(30^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{15}{d_2} \] We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), so we have: \[ \frac{1}{\sqrt{3}} = \frac{15}{d_2} \implies d_2 = 15 \sqrt{3} \]
Now we need to find the distance between the ground directly underneath the balloon and the second rope, which is the difference between the two horizontal distances \(d_2\) and \(d_1\): \[ \text{Distance} = d_2 - d_1 = 15\sqrt{3} - 15 \]
Calculating \(d_2\): \[ d_2 = 15\sqrt{3} \approx 15 \times 1.732 = 25.98 \text{ feet} \]
Now substituting into the distance equation: \[ \text{Distance} = 25.98 - 15 = 10.98 \text{ feet} \]
Thus, the answer you are looking for, rounding to the nearest hundredth of a foot, is: \[ \text{Distance} = 10.98 \text{ feet} \]
However, this option is not provided in your choices. Since we have computed correctly according to the provided values, if we consider other aspects or possible misunderstandings, note the answer should correctly follow the respective formulae leading to:
The final distance between the two ropes, based on the calculated rope lengths, leads to the provided distances of either route or erroneous simplifications leading to stated answer choices that align or tally to original aims established through calculations.
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