Tracking a dive A biologist standing at the bottom of an 80-fo...

Question

Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure).

a. Express the angle of elevation θ from the biologist to the falcon as a function of the height h of the bird above the ground. (Hint: The vertical distance between the top of the cliff and the falcon is 80−h.)

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says a woman stands on flat ground at the base of a 90-foot observation tower. A hawk takes off from the top of the tower, descending at an angle of 45 degrees below the horizontal. Below the statement, we're given a diagram of what was described in the statement. Now we're asked to write an expression for the angle of elevation data from the woman to the hawk in terms of the current height, H above the ground, and we're given a hint that the hawk has descended a vertical distance of 90 minus H from the top of the tower. So we need to determine our angle theta as a function of H. And we're gonna start off by using the hint that we were given in the problem. We're told that the hawk has descended a vertical distance of 90 minus H from the top of the tower. Now that vertical distance, along with the horizontal distance that the hawk has traversed, creates a right triangle, and The angle that was given for the angle of descent is 45 degrees, so that means that we actually have an isosceles right triangle, since two of the angles are 45 degrees. That means that the horizontal distance traversed by the hawk is also 90 minus H. And that horizontal distance also happens to be the leg of the right triangle that is adjacent to our angle of the. So, we'll label that as 90 minus H as well. And so now we can use trigonometry to find our angle theta. So recall that the tangent of data is equal to the opposite leg divided by the adjacent leg. So, in this case, we're gonna have the tangent of theta is equal to the opposite leg, which is the height of the hulk, which is H. That is divided by the adjacent leg, which we found was 90 minus H. So now I just need to take the inverse tangent of both sides. And we'll get theta is equal to the inverse tangent. Of the quantity of h divided by the quantity of 90 minus h. So this here is the answer that we're looking for. We determined our angle theta as a function of H. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Trigonometric Functions

Angle of Elevation

Vertical and Horizontal Distances

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