Giulia unreels her kite string until the kite is flying on a string of length 50 m. A light breeze holds the kite such that the
string makes an angle of 60° with the ground. After a few minutes, the wind picks up speed. The wind now pushes the kite until the string makes an angle of 45° with the ground. What distance has the kite changed position horizontally over the ground. Determine an exact expression.
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You can show that by constructing a triangle.
Take two points, O(0, 0) and A(1, 0), and let B be the point on the unit circle such that the angle between the line segments OA and OB is radians.
Since both A and B lie on the circle, the line segments OA and OB both have length 1 (same as the circle's radius). We finish constructing the triangle by connect A and B.
Since OB and OA have the same length, triangle OAB is isosceles, but more than that, it's also equilateral. Why? Because the interior angles of any triangle always add to radians. We know one of the angles is
radians, which leaves a contribution of
radians between the remaining angles A and B. Angles A and B must be congruent (because OAB is isosceles), which means they also have measure
radians.
Next, draw an altitude of the triangle through point B, and label the point where it meets the "base" OA, C. Since OAB is equilateral, the altitude BC is also a perpendicular bisector. That means OC has length , and by definition of
we have
Given:
Total number of cubes = 40
Red cubes = 18
Blue cubes = 13
Yellow cubes = 9
Aysha adds some more red cubes into the bag.
The probability now that she will take a red cube is 0.5.
To find:
The probability that she will take a yellow cube.
Solution:
Let Aysha adds x red cubes into the bag. Then,
Number of total cubes = 40+x
Number of red cubes = 18+x
Probability of getting a red cube is:
The probability now that she will take a red cube is 0.5. So,
Divide both sides by 0.5.
It means Aysha adds 4 more red cubes into the bag.
Now, the total number of cubes int he bag is:
Probability of getting a yellow cube is:
Therefore, the required probability is .