Two rays with common endpoint O form a 30 degree angle. Point A lies on one ray, point B on the other ray, and AB=1. What is the maximum possible length of OB?
If we call the lengths OA and OB x and y, then we have, using the law of cosines,
x^2+y^2-√3 xy = 1
That is just an ellipse with its extreme values at x=2 or y=2.
So, the maximum value of OB is 2.
To find the maximum possible length of OB, we need to draw a diagram and analyze the problem.
Step 1: Draw a horizontal line to represent one of the rays with common endpoint O.
Step 2: Measure 30 degrees counterclockwise from the initial ray and draw another ray with common endpoint O.
Step 3: Label the point where the second ray intersects the horizontal line as point A.
Step 4: Measure 1 unit from point A along the second ray and label that point as B.
Step 5: Draw a vertical line from point B to the horizontal line, labeling the intersection point as C.
Step 6: Observe that triangle OBC is a right triangle with angle BOC measuring 90 degrees.
Step 7: Use trigonometry to solve for the length of OB.
- Let x be the length of OB.
- Using trigonometry, we can say that:
- cos(30 degrees) = BC / OB
- cos(30 degrees) = 1 / x
- x * cos(30 degrees) = 1
- x = 1 / cos(30 degrees)
- Calculate the value of x using a calculator:
- x ≈ 1.155
Therefore, the maximum possible length of OB is approximately 1.155.
To find the maximum possible length of OB, we need to determine the position of points A and B that would result in the longest length for OB.
Since point A lies on one ray and point B lies on the other ray, we can consider the two rays to be the arms of an angle.
To maximize the length of OB, we need to position point A as close as possible to the vertex O of the angle and point B as far away as possible from the vertex O.
Here's how we can find the maximum possible length of OB:
1. Place point A close to the vertex O of the angle. Let's say point A is very close to O, such that AO is almost zero.
2. Draw a line perpendicular to the ray that contains point B, going through point A.
3. The maximum possible length of OB will be the length of the line segment formed by point B and the intersection point between the perpendicular line (step 2) and the ray containing point B.
4. To find the intersection point, we can use the fact that perpendicular lines form right angles. Since we know that the angle formed by the two rays is 30 degrees, the perpendicular line will form a right angle with one of the rays. Therefore, we can draw a right triangle with the perpendicular line as the hypotenuse and one of the rays as one of the legs. The angle opposite the leg will also be 30 degrees.
5. Now, we can use trigonometry to find the length of the line segment between the intersection point and point B. We know that AB = 1 and the angle opposite AB is 30 degrees. We can use the trigonometric definition of sine (sin) to find the length of the line segment OB.
sin(30°) = Opposite / Hypotenuse
sin(30°) = OB / 1
OB = sin(30°)
6. Calculate the value of sin(30°) using a calculator or by referring to a trigonometric table. The value of sin(30°) is 0.5.
OB = 0.5
Therefore, the maximum possible length of OB is 0.5.