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MAXImum [283]
3 years ago
8

A palm tree is supported by two guy wires as shown in the diagram below. Which trig expression can be used to find the height on

the tree where the top guy wire attaches if the base of the wire is four feet from the base of the tree? A sin (30° + 45°) B cos (75° – 45°) C tan (75° – 45°) D tan (30° + 45°)

Mathematics
1 answer:
Volgvan3 years ago
5 0

Answer:

D. tan(30^\circ+45^\circ) is the correct answer.

Step-by-step explanation:

The given situation can be represented as a figure attached in answer area.

B is the base of tree.

C is the base of wires.

A and D are the end of 2 wires supporting the tree.

\angle DCB =45^\circ\\\angle ACB =75^\circ\\

Here, we need to find the Height of the tree which is represented the by side AB.

and we are given that bases of wires and tree base are at a distance 4 ft.

i.e. side BC = 4 ft

If we look at the \triangle ABC, we are given the base BC and the \angle ACB, and the perpendicular is to be find out.

We can use trigonometric identity:

tan\theta =\dfrac{Perpendicular}{Base}

tan 75^\circ = \dfrac{AB}{BC}\\\bold {tan (45+30)^\circ }= \dfrac{AB}{4}\\\Rightarrow AB = \bold {tan (45+30)^\circ } \times 4 = 14.93\ ft

Hence, D. tan(30^\circ+45^\circ) is the correct answer.

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We can solve this by setting up 2 equations. We can use any letters like. For now, I'll use x and y.
So now we know that    x - y = 0.7   and    x + y = 1.
Now we can eliminate one of the letters by adding or subtracting one equation from the other. I am going to eliminate y by adding the two together (the y and the -y cancel).
This gives us   2x = 1.7,    and so x = 1.7 ÷ 2 = 0.85
Finally, we can substitute x for 0.85 back into one of the original equations to figure out what y equals. I'm going to use   x + y = 1.
So now we have    0.85 + y = 1,    so y = 1 - 0.85 = 0.15

The numbers, therefore, are 0.85 and 0.15 (you can check by using the other equation).

Hope this helps!
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