Answer the following questions, showing your work at each step:

Question

1. You want to hang a 600 pound statue from your ceiling for a party. It will be hung by two cables each making a 60 degree angle with the ceiling. How much tension will be in each of the cables? Round your answer to the nearest pound.
2. Find all solutions for the equation of 3cos(t)+4=2 on the interval [0, π), or answer "N/A" if there is no solution.
3. Consider the numbers 0, 1, 2, 3, and 4. Take the square root of each of these numbers, then divide each by 2. Describe the relationship between the values you receive and the trigonometric ratios.
4. A Ferris wheel boarding platform is 4 meters above the ground, has a diameter of 66 meters, and makes one full rotation every 5 minutes. How many minutes of the ride are spent higher than 47 meters above the ground?

Answer 1

1. Since there are two cables, both at the same angle, we know for sure that the tension in both cables is the same. The vertical force component of each cable must therefore be half the weight. You can now think of the cables and their forces as two triangles with a base of 600lb/2=300lbs and the hypotenuse as the tension in the cable. If you this, then you can set up the equation Tsin(60deg)=300, and then solve for the tension, which comes out to be 300/0.866=346lbs.
2. First, you have to get it in the form cos(t)=constant, so subtract each side by 4 and then divide by 3. That leaves you with cos(t)=-2/3. Since this is not one of the big trig answers, you have to use a graphing calculator. When you use a graphing calculator, you find that there is only one solution, at 2.301 rad.
3. These are the values you get when you take the square roots and divide them by 2: 0/2 = 0, 1/2, √ 2/2, √ 3/2, and 2/2 = 1. The only thing I can see are that these values directly mimic the "special" or "normal" values for a sin function in the first quadrant: sin 0 = 0, sin 30 = 1/2, sin 45 = √2/2, sin 60 = √3/2, and sin 90 = 1.
4. I would solve this by simply drawing a circle with a radius of 33 meters on top of a 4 meter tall platform. Now, you can draw a dotted line through some part of the upper half of the Ferris wheel and label it 47 meters. Then, you can draw a right triangle with corners at the center, the edge of the circle where the dotted line goes through, and then under that point on the horizontal radius. This allows you to solve for the angle when the rider is at 47 meters. The height of the triangle is 47 meters, minus the radius and the platform, so 47-33-4 meters=10 meters. This means that our angle is arctan(10/33)=16.86 degrees. Assuming symmetry, our second angle at 47 meters is 180 degrees-16.86 degrees=163.14 degrees. This means that the total change of angle between the two points is 146.28degrees. Since it takes 5 minutes per rotation, the time spent over 47 meters is the ratio of the arc length over the full circle times 5 minutes, so it would be (146.28/360)*(5 min)=2.03 minutes.

Hope this helps!