A 3m high fence is on the side of a hill and tends to lean over. The hill is inclined at an angle of 20 degree to the horizontal. A 6.3 m brace is to be installed to prop up the fence. It will attached to the fence at a height of 2.5 m and will be staked downhill from the base of the fence. What angle, to the nearest degree, does the brace make with the hill?
the downhill angle of the fence is 110º ... 90º + 20º
using the law of sines ... sin(Θ) / 2.5 = sin(110º) / 6.3
Could you explain it in a little more depth.
thank you
To find the angle that the brace makes with the hill, we can create a right triangle using the hill, the fence, and the brace as the three sides.
Let's label the sides of the right triangle:
- The height of the fence is 3 m (let's call it side a).
- The distance from the fence to the base of the hill is 6.3 m (let's call it side b).
- The height from the attachment point on the fence to the top of the hill is 2.5 m (let's call it side c).
Now, we can use the trigonometric tangent function to find the angle:
tan(angle) = opposite/adjacent
Opposite side = side c - side a = 2.5 m - 3 m = -0.5 m (we use "-" because side c is below side a)
Adjacent side = side b = 6.3 m
tan(angle) = (-0.5 m) / (6.3 m)
angle ≈ tan^(-1)(-0.5/6.3)
Using a calculator or a math software:
angle ≈ -4.54 degrees
The angle that the brace makes with the hill, to the nearest degree, is approximately -5 degrees.
To find the angle that the brace makes with the hill, we can create a right triangle and use trigonometry.
Let's label the given information:
Height of the fence (opposite side): 3m
Distance from the fence to the base of the hill (adjacent side): unknown (let's call it x)
Height where the brace is attached (adjacent side): 2.5m
Length of the brace (hypotenuse): 6.3m
First, let's find the distance from the fence to the base of the hill (x). We can use cosine:
cos(angle) = adjacent/hypotenuse
cos(angle) = x/6.3
To isolate x, we can rearrange the equation:
x = 6.3 * cos(angle)
Next, let's find the angle that the brace makes with the hill. We can use the tangent function:
tan(angle) = opposite/adjacent
tan(angle) = 3/x
Again, to isolate the angle, rearrange the equation:
angle = arctan(3/x)
Now, substitute the value of x:
angle = arctan(3/(6.3 * cos(angle)))
To find the angle to the nearest degree, we can use a numerical method or approximation technique. One approach is to use the bisection method, where we keep approximating the angle until we get a close enough value.
Let's start with an initial angle, say 30 degrees:
angle = arctan(3/(6.3 * cos(30)))
Using a calculator, compute the value of the right side of the equation. If the value is close to 30 degrees, then it is our answer. Otherwise, adjust the angle and repeat the process until the desired degree of accuracy is achieved.
Note: Calculating the angle precisely using the method described above can be complex and time-consuming. Approximation techniques like the bisection method or utilizing numerical methods in programming can be helpful. Alternatively, using specialized software or online tools can provide a more straightforward and accurate solution.