Precalculus Question: Are my answers for the following problems correct?

Question

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.

My Answer

If 2 cables are holding up 600 pounds of force, 1 cable must hold up 300 (600/2 = 300). From there, we can construct two triangles using the 60 degree angle, the tension T as the opposite side, and 300 as the adjacent side. Our equation is now Tsin(60degrees)=300. Solving for this, we get 346.420 pounds, which works out to be 346 pounds (rounded down).

Question 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. You may not use a graphing utility and must solve showing your work in algebraic form.

My Answer

Start off by simplifying the equation, so 3cos(t)+4 = 2 becomes 3cos(t) = -2, then cos(t) = -2/3. Because the interval is (0, π) our answer must be less than π radians. If cos(t) is -2/3, we need to find cos inverse, so cos^-1(-2/3) = 2.300 radians. This can be converted into 0.73π radians.

Question 3: What is the relationship between the square root of 0, 1, 2, 3, and 4 divided by 2 and the trigonometric ratios? (I believe this is subjective, but I am not sure.)

My Answer

sqrt(0) = 0; 0/2 = 0

sqrt(1) = 1; sqrt(1)/2 = 0.5

sqrt(2) = 1.414... ; sqrt(2)/2 = 0.707...

sqrt(3) = 1.732; sqrt(3)/2 = 0.866...

sqrt(4) = 2; sqrt(4)/2 = 1

These mimic the sine function in its first quadrant. That is, sin(0deg) = 0, sin(30deg) = 0.5, sin(45deg) = 0.707..., sin(60deg) = 0.866, and sin(90deg) = 1.

Question 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?

My Answer

Amplitude (a): radius (1/2 diameter) = 66/2 = 33

Period (g): Length of one cycle (in this case, 5 minutes)

Vertical displacement (h): 4 + 33 = 37

Assume φ is the period, or 2pi/g = 2pi/5. Thus we have the following equations to solve for t:

h(t) = asin(gt)+h = 33sin(2pi/5t) + 37

h(t) = acos(gt)+h = 33cos(2pi/5t) + 37

Now, set these equations equal to the height above ground (47)

47 = 33sin(2pi/5t) + 37 = 33cos(2pi/5t) + 37

10 = 33sin(2pi/5t) = 33cos(2pi/5t)

10/33 = sin(2pi/5t) = cos(2pi/5t)

2pi/5t = sin^-1(10/33) = cos^=1(10/33)

t = 5/2pi*sin^-1(10/33) = 5/2pi * cos^-1(10/33)

Now find the two differences in times.

t = t2-t1

= 5/2pi(pi-sin^-1(10/33))-5/2pi*sin^-1(10/33)

= 2.0100...

t = t2-t1

= 5/2pi*cos^-1(10/33)-(-5/2pi*cos^-1(10/33))

= 2.0100...

Answer 1

Let me say at the start that if I were grading your paper, I would commend you for taking the time to explain your steps in solving, and for filling in some details (making observations) that were not required. Well done! However, there are a few things to look at a second time.

1) Tension = Force acting on the string = mass x acceleration due to gravity.

You are correct that because the angles are the same, the tension in the two ropes will be equal.

The Physics involved here is confusing. Is weight of the statue = force?

Is 300 pounds a measure of mass? or force acting on an object?

According to the literature, a "pound" is not a measure of mass but a measure on the force acting on an object on Earth. Were we to move the statue to a planet with 1/2 the gravity of Earth, we'd need to multiply 300 x 1/2 = 150 to get the "right hand side" of the equation you are solving.

Good job!

By the way, did you know that the "imperial" measure of mass is a slug? Our statue's mass is about 18.6 slugs. Could you imagine going to your family physician and being told you are 5.6 slugs in mass? (Maybe the metric system isn't such a bad idea after all.)

2) Correct! Here you were "lucky" if that is a word, that the problem involved cosine, and that the instructions specifically asked for angle(s) in the (0,pi) interval. No need to use reference angles to find a solution. You even went so far as to express the answer as a decimal, AND as a decimal multiple of pi.

Well done! Extra credit there.

3) People use this pattern as a memory device (mnemonic device) to help with memorizing trig values of special angles, starting with sine and extending to cosine, then tangent, and all six functions. Then extending to all angles between 0 and 2pi.

But you are essentially correct. Well done.

4) This was a complicated problem. You did well. At first I thought you had forgotten to account for the 4 meter high platform, but I see you did that at the beginning. I would only ask that you double check something in this last problem. Make sure you have the amount of time ABOVE 47 feet, and not accidentally the amount of time BELOW 47 feet. Either it's 2 minutes above and 3 minutes below or the other way around. How can you verify that? Also you have to make a big assumption: that the ferris wheel is moving at a constant rotational speed. It's worth noting that in your answer. Stating things like this show that you really understand the physical realities of the situation/problem being posed.

Again, nice work!