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

Question

Right triangle with height x, base 1600, angles 66° and 24° for solving length x.

Accepted Answer

Hello, everyone. We are asked to determine the value of X rounded to the nearest integer. We were given a right triangle within the right triangle is a smaller right triangle. Both share a adjacent side with the given angle that is 1600. We have the smaller right triangle that has an angle of 24 degrees and the side opposite it is not labeled. Then our larger rate triangle has an angle of 66 degrees and the side opposite, it has a section labeled X. But another section that is unlabeled, our answer choices are a units, B 4036 units C 4306 units, D 2881 units. So beginning by looking at my smaller triangle, the one that has 24 degrees as an angle, the side across from it or opposite, it is unlabeled. I'm going to label it, I'm gonna call it D and now I'm going to use the information I have to find the value of D I have an angle. I have a variable opposite it. And I know the site adjacent to it. So I'm gonna work with tangent. So we're gonna say the tangent of 24 degrees equals D divided by using a calculator. I find the approximate value of the tangent of 24 degrees to be 0.4452. And that still equals D divided by to get D alone. I di I multiply both sides by 1600. And when I do, we find that D is approximately 712.365 units. So I'm gonna label that on my image 712 point 3659. I am doing a little rounding but I am not rounding all the way to the nearest integer just yet. So now that I know a little bit more about the side opposite our angles, I'm going back to my larger right triangle. I could say the site opposite the angle 66 degrees is X plus D and now I have a value for D. So I'm gonna work with the tangent of degrees which is the site opposite it. So X plus D so X plus 712. divided by the site adjacent to it, which is 1600. I'm gonna use a calculator to find the approximate value of the tangent of degrees. And I find that that is 2. and that's still equal to my fraction X plus 712.3659 divided by 1600 to try to isolate the variable and multiplying both sides by 1600. And when I do, we get 3593. equals X plus 712. to get the X by itself. I'm gonna subtract the 712.3659 from both sides. And when I subtract that we get 2881. equals X. We went around to the nearest unit. So X will equal units. And that is answer choice D 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 Relationships in Triangles

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