Answer:
1) Therefore, the height of the waterfall above the observer is approximately 688.62 feet.
2) Therefore, the length of the arc intercepted by a central angle of I radians in a sector of a circular patio with a diameter of 24 feet is 12I feet.
Explanation:
1)
1. Draw the Triangle ABC
2. point A represents the top of the waterfall, point B represents the far side of the pool, and point C represents the observer's position. We want to find the value of x.We know that the angle of elevation from point C to point A is 75°, so we can use the tangent function to relate the angle to the opposite and adjacent sides of the triangle
tan(75*)= opposite/adjacent
3. In this case, the opposite side is x (the height of the waterfall above the observer), and the adjacent side is the distance from the observer to the far side of the pool, which is given as 185 feet. So we can write:
tan (75*)= x/185
4. To solve for x, we can multiply both sides by 185:
x= 185* tan(75*)
5. Using a calculator, we can evaluate tan(75°) to be approximately 3.732, so:
x=185*3,732
6. The solution is 688.62 Feet
2)
1. The length of the arc intercepted by a central angle is given by the formula
length of arc = (central angle / 2π) * circumference of circle
2. n this case, the circular patio has a diameter of 24 feet, which means the radius is 12 feet. So the circumference of the circle is:
circumference = 2π * radius
circumference = 2π * 12
circumference = 24π
3. Now we need to find the length of the arc intercepted by a central angle of I radians. We can substitute these values into the formula:
length of arc = (I / 2π) * 24π length of arc = I * 12
4.So the length of the arc intercepted by the central angle is 12 times the measure of the angle in radians.
5.Therefore, the length of the arc intercepted by a central angle of I radians in a sector of a circular patio with a diameter of 24 feet is 12I feet.