Pre-Calculus chapter 6.1

Question

A Ferris wheel is 45 meters in diameter and boarded from a platform that is 2 meters above the ground. The six o'clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 6 minutes. How many minutes of the ride are spent higher than 28 meters above the ground?

Answer 1

Hi Mira -

Thanks for your question. To solve this one, we need to generate a function that represents the rider's height above the ground, and then look for values where that function has an output greater than 28 - if we can figure out the x-value at which it goes above 28 and then the x-value at which it dips back below 28, the difference between those 2 x-values will be the total time above 28 meters.

Depending on where you're at in this unit in class, you may have learned that circular motion of this sort is easiest to represent using sine and/or cosine functions. If this isn't something you're familiar with, let me know and I'd be happy to do a session to explain a bit more about why this is.

Because our ferris wheel rider starts at the minimum height of 2 meters, it makes sense to use a cosine function rather than a sine function, because cosine starts at either the maximum or minimum rather than in the middle.

We will use the form y = a cos (bx) + c to achieve the necessary transformations of a regular y = cos x graph - a will set our amplitude, b will set our period, and c will set our vertical shift.

Amplitude: To find the amplitude of a sine or cosine function, we take half the difference between the highest and lowest y-values it will take on. In this case, our minimum height is 2 meters, and our maximum height will be 47 meters (since the diameter of the ferris wheel is 45 meters and the bottom is at a height of 2 meters). To calculate the amplitude, we do the following:

(47 - 2)/2 = 22.5

Finally, because our cosine graph is unusual in that it needs to start out at its minimum rather than the normal starting location of its maximum (in other words, it's upside down), our amplitude (and our value of a in the equation) needs to change signs to create this vertical flip, so it will be -22.5 rather than +22.5.

Period: In a word problem, you can generally tell what the period of a trig function to model the situation should be from the part of the problem that describes the amount of time it takes to complete one cycle of a repetitive process. In this case, the repetitive process we are talking about is laps around the ferris wheel, and we're told that one full revolution takes 6 minutes, so the period of our cosine function will need to be 6. In order to make a sine or cosine function have a particular period, we use this equation:

period = 2π/b

where b is the value of the coefficient in the general equation I wrote at the top. In this case, the period needs to be 6, so we can plug that in for the period and solve for the correct value of b:

6 = 2π/b

6b = 2π

b = (2π)/6

b = π/3

Vertical Shift: The vertical shift (value of c) for a transformed sine or cosine function will always be equal to the midline of the graph - that is, the y-value halfway between the highest and lowest y-values. All we need to do here is find the y-value halfway between the minimum height (2) and the maximum height (47) by averaging those two numbers:

c = (2 + 47)/2 = 49/2 = 24.5

Now that we have our values for a, b, and c, we can fill in the coefficients in our function:

f(x) = -22.5 cos ((π/3) * x) + 24.5

At this point, I'd imagine your teacher expects you to turn to a calculator or computer graphing system to figure out where that function will have y-values greater than 28. It's also possible to set the function equal to 28 and solve algebraically for x, but you're not going to be able to simplify the resulting expression very much without a calculator anyway. If it's explicitly stated somewhere that this is intended to be a non-calculator problem, let me know and we can talk more about how to do it without a calculator.

I'd recommend plugging this function into either your graphing calculator or desmos, and then also graphing the line y = 28 on the same set of axes in order to make it really clear where the cosine graph is above or below 28. Using your graphing calculator's intersect function, or just by clicking on the intersection points on desmos, we can see that the height first goes above 28 at x = 1.649 minutes, and dips back below 28 at x = 4.351 minutes. Subtracting these values, we get:

4.351 - 1.649 = 2.702

Therefore, we know that the ride is higher than 28 meters above the ground for 2.702 minutes. I hope this is helpful! Let me know if you have any other questions and I'll be happy to explain more, and if you'd like to give a session with me a try to work on more problems like this, just send me a message and we can set something up soon!