Final answer:
The length of the rope is approximately 20 feet, and the distance from the base of the pole to the spot where the rope is fixed to the ground is approximately 17.32 feet.
Step-by-step explanation:
To find the length of the support rope, we can use trigonometry. In this case, the rope forms a right triangle with the pole and the ground. We know that the angle between the rope and the ground is 30°, and we know the height of the pole is 10 feet. Using the sine function, we can set up the equation:
sin(30°) = opposite/hypotenuse
sin(30°) = height of pole/length of rope
Substituting the values we have, we get:
sin(30°) = 10/length of rope
Solving for the length of the rope, we get:
length of rope = 10/sin(30°)
Calculating this, we find that the length of the rope is approximately 20 feet.
To find the distance from the base of the pole to the spot where the rope is fixed to the ground, we can use trigonometry again. In this case, we have a right triangle formed by the distance from the base to the spot, the height of the pole, and the length of the rope. The angle between the ground and the rope is 30°, so we can use the cosine function to set up the equation:
cos(30°) = adjacent/hypotenuse
cos(30°) = distance from base to spot/length of rope
Substituting the values we have, we get:
cos(30°) = distance from base to spot/20
Solving for the distance from the base to the spot, we get:
distance from base to spot = 20 * cos(30°)
Calculating this, we find that the distance from the base of the pole to the spot where the rope is fixed to the ground is approximately 17.32 feet.