In Exercises 29–36, find the length x to the nearest whole unit.

Question

Right triangle with angles 22° and 54°, base 200 units, height x to find.

Accepted Answer

Hello, everyone. We are asked to determine the value of X rounded to the nearest integer. We are given a large rate triangle with one of the legs measuring X within the larger triangle is a smaller right triangle with an angle of 54 degrees. The larger right triangle has an angle of 22 degrees and a section of the base for the larger right triangle measures 200. Our answer choices are a 114 units, B 141 units C 62 units D 26 units. So first, I'm gonna think about what do I know? What? Don't I know, I don't know the space of the smaller right triangle. So I'm gonna give that a value. I'm calling it D just as a placeholder because now I can say if I look at my smaller right triangle, I know I have an angle of 54 degrees and X is opposite it D is adjacent to it. So if I take the tangent of 54 degrees, that'll equal my opposite side, X divided by my adjacent side D. Now looking at my larger right triangle, I can again use tangent now be the tangent of degrees. The site opposite is still X, the site adjacent, however, is now the length 200 plus D. So I now have the tangent of 54 degrees equals X divided by D and the tingent of 22 degrees equals X divided by in parentheses, 200 plus D. I'm going to find what the tangent of these angles are. And then I'm gonna use algebra to set these two equations equal to each other. The tangent of 54 degrees is approximately 1.3764. So that's gonna equal X divided by D and the tangent of 22 degrees is approximately 0.40403. And that is still equal to X divided by 200 plus D in parentheses. I'm gonna solve each of these equations for X. And when I do so I have 1.3764 D equals X. And in my other equation, I have 0.40403, multiplied by the parentheses, 200 plus D, close parentheses equals X. Now that I have both equations equal to X, I can set them equal to each other. So I will have 1.3764, D equal two. And I'm gonna distribute into my parentheses in this step. So it's gonna be equal to 80. plus 0. 3d. So they're not the prettiest of numbers. But if we remember our algebra rules, we can do this. I'm gonna move all the DS to one side. So I'm subtracting 0.4040 3d from both sides. And when I do so I get 0.97237, D equals 80.806, dividing both sides by 0.97237. I find that D equals approximately 83.1021. Now you're like, that's nice. But we wanted X. All right. Well, now that we know D we can plug it back into one of those equations and find X. So what I'm gonna do is I'm gonna plug the value of D into the equation 1.3764 multiplied by D. So 83.1021 and that equaled X. And when I multiply those two things together, I get 114.38 equals X and rounding to the nearest integer, we get answer choice. A 114 units. Have a nice day.

In Exercises 29–36, find the length x to the nearest whole unit.

Key Concepts

Right Triangle Properties

Trigonometric Ratios

Angle of Elevation

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