vertical pole 20 feet high is sighted horizontally from a point 75 feet from its base up the sloping side site of a hill on which it rest.what is the inclination of the slope?
Did you draw a diagram? If so, then it's easy to see that
sinθ = 20/75
Well, let's see. You have a 20-foot tall vertical pole and you're sighting it horizontally from a point 75 feet away on the side of a slope. So basically, we're trying to find out how steep that slope is, right?
You can think of it this way: the pole forms one side of a right triangle, with the slope forming the other side. The distance from the base of the pole to the point where you're standing is the hypotenuse.
Now, we can use the Pythagorean theorem to find the length of the slope. If we call the length of the slope "x," we can say that x^2 + 20^2 = 75^2. Solving for x, we get x ≈ 74.15 feet.
So the slope is about 74.15 feet long. Now, to find the inclination of the slope, we can use the tangent function: tan(inclination) = 20/74.15. Taking the arctan of both sides, we find that the inclination of the slope is approximately 15.4 degrees.
So, the inclination of the slope is around 15.4 degrees. I hope that answers your question... and I hope my math didn't take you on too much of a slope-tastic journey!
To find the inclination of the slope, we can use trigonometry. We can create a right-angled triangle with the vertical pole, the distance from the base of the pole to the viewing point, and the height of the hill.
The length of the adjacent side of the triangle (the base of the hill) is 75 feet, and the length of the opposite side of the triangle (the height of the hill) is 20 feet.
The tangent of the angle of inclination is equal to the opposite side divided by the adjacent side. So, the tangent of the angle is 20/75.
We can use the inverse tangent function (tan⁻¹) to find the angle:
Angle of inclination = tan⁻¹(20/75)
Calculating the inverse tangent of 20/75, we find that the angle of inclination is approximately 15.64 degrees.
Therefore, the inclination of the slope is approximately 15.64 degrees.
To find the inclination of the slope, we can use trigonometry. Let's start by drawing a diagram to visualize the situation.
We have a vertical pole 20 feet high and a distance of 75 feet from the pole's base up the sloping side of the hill. This forms a right triangle with the pole as the height, the distance up the hill as the base, and the slope of the hill as the hypotenuse.
Now, let's use the properties of right triangles and trigonometry. The tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side.
In our case, the opposite side represents the height of the pole, which is 20 feet, and the adjacent side represents the distance up the hill, which is 75 feet. We want to find the angle, which is the inclination of the slope.
So, we can use the formula:
tan(θ) = opposite / adjacent
Substituting in the values we know:
tan(θ) = 20 / 75
Now, to find the angle θ, we can take the inverse tangent (also known as arctan or tan^(-1)) of both sides:
θ = tan^(-1)(20 / 75)
Using a calculator or a trigonometric table, we can determine the value of tan^(-1)(20 / 75) to find the inclination of the slope.