Solve each problem. (Source for Exercises 49 and 50: Parker, M., ...

Question

Solve each problem. (Source for Exercises 49 and 50: Parker, M., Editor, She Does Math, Mathematical Association of America.)

Height of a Tower The angle of depression from a television tower to a point on the ground 36.0 m from the bottom of the tower is 29.5°. Find the height of the tower.

Accepted Answer

Hello, today we're going to be finding the height of the pole. If the angle of depression from the top of the pole to point A in the garden is 34.5 degrees point A is 24 m away from the bottom of the pole. So if we take a look at the given image, the image with the given information can be used to form a right triangle. Our goal is we want to find the height of the pole which is line B to C. And since we don't know this height, we can label the side of the right triangle with the variable X. We are also told that the distance from the bottom of the, from the bottom of the pole to point A is 24 m. So how can we use this information to solve for the side of X? Well, if we take a look at angle A, we can label the angle A as an angle theta, we want to find the opposite side and we know the distance to the adjacent side of the right triangle. So why don't we go ahead and use the tangent function to help us find the height of the pole. So for example, recall that tangent of theta is equal to Y divided by X. In this case, here Y is going to represent the height of the pole and X is going to represent the length or the adjacent side to the angle. Theta. If we plug in the known information, we can write tangent data is equal to the height of the pole which is currently X divided by the adjacent side or the length from the bottom of the pole to point A which is 24 m. If we solve for X in this given equation, by multiplying both sides of the equation by 24 we can write that X is equal to 24 multiplied by tangent of the. So if we want to solve for X, we're going to need to solve for the angle theta which is going to be angle A well, we can solve for angle A by finding another angle within the right triangle. Recall that the angle of depression from the top of the pole is given to us as 34.5 degrees. Now, if we use the vertical line at the very top of the pole with the height of the pole, we can form a right angle. This right angle has a total of 90 degrees. So if we want to find the angle inside of the right triangle, we can take 90 degrees and subtract 34.5 degrees to get the angle B inside of the right triangle. So angle B is equal to degrees minus 34.5 degrees. 9, 90 minus 34.5 will leave us with 55.5 degrees. So that means the angle B inside of the right triangle is 55.5 degrees. Now again, angle C is given to us as a right triangle which is 90 degrees. Now that we know the measurement of two of the angles inside the triangle, we can use the angle sum equation to solve for the angle. Theta the angle sum equation for a right triangle states that the sum of all the angles inside the triangle. So in in this case, angle B plus angle C plus angle A will equal to 180 degrees. We know that angle B is 55.5 degrees. We know that angle C is 90 degrees and we know that angle A is being represented with data. So we can use this equation to solve for the 55.5 plus 90 will give us a value of 145.5. So we have 145.5 plus data is equal to 180. Finally, in order to solve for data, we can sub, we can subtract 145.5 to both sides of the equation. This will leave us with data is equal to 34. degrees. So what this means is that angle A has a total degree of 34.5. And now that we know that angle theta or angle A is 34.5 degrees, we can substitute theta in our equation for X in order to solve for the height of the pole. So this will give us X is equal to multiplied by tangent of 34.5 degrees. Now, at this point, I would recommend plugging in this value into a calculator. And when doing so, we're going to get a continuous decimal. This decimal will be 16.494747. And continuing, we're going to round this decimal to one decimal place. So we're going to take a look at the number to the right of four, which is nine. Since this value is greater than five, we're going to round four up one unit and that means the height of the pole X will equal to 16.5 m. This is going to be the answer to this question. And with that being said, the overall solution is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Solve each problem. (Source for Exercises 49 and 50: Parker, M., Editor, She Does Math, Mathematical Association of America.) Height of a Tower The angle of depression from a television tower to a point on the ground 36.0 m from the bottom of the tower is 29.5°. Find the height of the tower.

Key Concepts

Angle of Depression

Trigonometric Ratios

Right Triangle Properties

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