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

Question

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

Right triangle with angles 37° and 72°, height 800, find length x.

Accepted Answer

Hello, everyone. We are asked to determine the value of X rounding to the nearest integer. We are given a right triangle. And within that right triangle, we have a smaller right triangle. Both rate triangles share a side that is labeled 800. The smaller interior right triangle has an angle of 72 degrees. That is opposite side 800. The larger right triangle has an angle of 37 degrees. That is opposite side 800. And the base between the two angles is measured X. We have four answer choices. A 208 units. B 2213 units C 802 units, D 1322 units. So I'm sure there's a few different ways that you could do this. I'm gonna do the way that makes the most sense to me. And the first thing I'm gonna do is I'm gonna label the other half of my base with a variable. I'm gonna call it D just so it won't get confused with the other section that's labeled X. And then I wanna find what is the value of D? And I'm gonna use D to find X. So starting by finding the value of D, I know I have an opposite side and an adjacent side of my small right triangle. So opposite and adjacent will give me tangent. So the tangent of 72 degrees is the opposite side. So the measure of 800 divided by the adjacent side, which is D using a calculator, I find that the tangent of 72 degrees is approximately 3. and that'll equal 800 divided by D. And if I multiply both sides by D, I get 3.777, D was dividing both sides by 3.777. I find that D equals approximately 259. units. I'm not rounding all the way to an integer yet just to try to keep my numbers a little bit more precise, but I am gonna fill in my figure that D equals 259.9344. Now, I can think about my larger right triangle here. My base or the adjacent side will now be X plus D. So X plus 259. and the opposite side is still 800. So still having an opposite and an adjacent I'm again going to use the tangent function. So the tangent of 37 degrees equals my opposite side of 800 divided by my adjacent side, which is X plus 259.9344. I'm again going to use a calculator to find the tangent of 37 degrees and that comes out to be approximately 0.7536. And that will equal 800 divided by X plus 259.9344. So I'm gonna multiply both sides or cross multiply whichever way you want to think of it so that I no longer have a fraction. So I will have 0.7536, multiplied by the parentheses. X plus 259.9344, close parentheses. And that'll equal 800. Distributing into my parentheses. I have 0.7536, X plus 195. equals 800. And now I'm back to my normal steps for algebra. I'm gonna solve for X by subtracting the constant which is 195.8866 from both sides. I get 0.7536, X equals 604.1137. And I am rounding a little bit. I'm just not rounding all the way to the nearest integer just yet dividing both sides by 0.7536. I get a value for X that is 801.637. And if I want to round to the nearest integer, this would be 802 units. And that's our value for X. So X is 802 units which is answer choice C have a nice day.

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

Key Concepts

Right Triangle Trigonometry

Trigonometric Ratios (Sine, Cosine, Tangent)

Angle Sum in a Triangle and Complementary Angles

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