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Answer:
114.02 meters
Step-by-step explanation:
The information you're given about the tree will form a triangle, with the three corners being the top of the tree, the point where the tree meets the ground, and the specified point 41.5 meters from the tree. The angle of elevation (70°) is the angle made where the line representing the ground and the line connecting the top of the tree to the ground at the specified point connect.
The angle where the tree meets the ground is 90°, so the triangle a right triangle.
So this all means you know one angle of a right triangle (70°) and the side adjacent to this angle (41.5 meters). You're solving for the height of the tree, which is the side opposite of the angle you know. You can solve using either <em>Sine, Cosine, </em>or <em>Tangent.</em>
Use the acronym SOHCAHTOA to figure out which of those three functions to use. Feel free to look this up for more information if you haven't heard of it before. In this case, we're using Tangent (TOA = Tangent, Opposite side, Adjacent side)
To solve using any of those three functions, you'll need a scientific calculator. On your calculator, you'll see the buttons <em>SIN, COS, </em>and<em>TAN.</em> For this problem, press <em>TAN</em> and you'll see something appear that says <em>tan (___)</em>. You'll need to input a number to get your answer, and here's how to find that number:
Again referencing SOHCAHTOA, the "TOA" part lists Opposite first and then Adjacent. This is because you're going to write a fraction with those two numbers, and since Opposite is first, it's the numerator. That means that Adjacent is the denominator. We don't know the value of the opposite side, so it'll be represented by <em>x</em>, and we do know the value of the adjacent side, 41.5. This means the fraction is .
This isn't the number to input into the calculator -- you need to isolate the variable:
The original equation is:
tan 70 = x/41.5
Multiply both sides by 41.5 (To cancel out the denominator and isolate <em>x</em>)
tan 70 (41.5) = x
Now you've got the equation for <em>x, </em>you're ready to put this in your calculator.
What appeared on your calculator was:
tan(__)
Input like follows:
(tan(70))41.5
The brackets are needed like that so the calculator knows to multiply the entire thing by 41.5. You might get a wrong answer otherwise.
Hit enter, and you'l get the height of the tree:
114.02 meters
We have a circle of radius 5 m inside a square of side 10 m. The shaded area is:
Now, we calculate the area of the square and the circle:
Finally, the shaded area is: