Two wires stretch from the top T of a vertical pole to points ...

Question

Two wires stretch from the top T of a vertical pole to points B and C on the ground, where C is 10 m closer to the base of the pole than is B. If wire BT makes an angle of 35° with the horizontal and wire CT makes an angle of 50° with the horizontal, how high is the pole?

Accepted Answer

Welcome back, everyone. A flagpole at a schoolyard extends cables to points G and H on the flat ground, where H is 7 m closer to the base of the flagpole than G. The cable to. G forms an angle of 40 degrees with the horizontal, and the cable to point H forms an angle of 65 degrees with the horizontal. What is the height of the flagpole round your answer to two decimal places. We're given for answer choices A 6.52 m, B 7.38 m, C 8.22 m, and D 9.65 m. So let's draw two triangles. Let's suppose that our flagpole has a height of age. Now for the first triangle we're going to label our base angle as o. We're going to treat it as a right triangle, and we are going to consider the point G, right? Let's suppose that the side length OG is X meters. We know the angle is 40 degrees. Now for the second triangle, we're going to consider the point H, right? So now we're adding H. We still have the same flagpole, so we're going to have the same height H. and in this case, our angle is 65 degrees. And we also know that H is 7 m closer to the base of the flagpole than G, so our O H is simply X minus 7 in this case. And now we have those two triangles we can express H in terms of each angle. Let's recall that tangent is the ratio between the opposite side and the adjacent side. So our first equation is H divided by X is equal to tangent. Of 40 degrees and then we can say that age. Divided by x minus 7 is equal to tangent. Off 65 degrees. From each equation we can express H. So if we take our first equation, we can multiply both sides by X, which is not equal to 0. Therefore, H is equal to X multiplied by tangent. Of 40 degrees and now from the second equation. We multiply both sides by X minus 7, which is not equal to 0. So H is equal to x minus 7, multiplied by tangent. Of 65 degrees. And because we're expressing the same height, we can just equate those two, right? We can say that. Acts Multiplied by a tangent of 40 degrees is equal to now we can distribute and we can say that this is x tangent 65 degrees minus 7 tangent of 65 degrees. So we're going to equate those x tangent of 40 degrees is equal to x. Ancient Of 65 degrees minus 7. Ancient Of 65 degrees. And now what we want to do is simply solve for X. We can rearrange this equation, right? We can just say that X. Tangent of 40 degrees. Now we're going to subtract x tangent of 65 degrees from both sides, and this leaves us -7. Tangent of 65 degrees. So now what we're going to do is simply factor out the X and this leaves us with X, tangent 40 degrees minus tangents. 65 degrees, and this is equal to -7. Tangent of 65 degrees. So now what we're going to do is simply solve for X. X is basically the ratio between the right hand side, which is -7. Tangent of 65 degrees and our factor which is the difference of the two tensions so tangent 40 degrees minus tensions. Of 65 degrees. This is our solution for X, and essentially we want to solve for H. So we can use any of the two equations. Let's use the first one. We know that H is equal to X multiplied by a tangent of 40 degrees. So we can just rewrite our expression for X, right? And we can multiply it by tangent. Of 40 degrees and that's it we have our final expression. All that we have to do is simply calculate the result. So if we calculate the result. In round to two decimal places, we're going to get approximately 9.65 m, which is our final answer, and it corresponds to the answer choice D. Thank you for watching.

Two wires stretch from the top T of a vertical pole to points B and C on the ground, where C is 10 m closer to the base of the pole than is B. If wire BT makes an angle of 35° with the horizontal and wire CT makes an angle of 50° with the horizontal, how high is the pole?

Key Concepts

Trigonometric Ratios

Right Triangle Properties

Angle of Elevation

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