Pls help, sorry for the trouble

Question

Link to image of the problem  

pls help with part B, and if you could pls check if my answers are correct for part A

Answer 1

Hi Lia, it is no trouble at all. Let's see if we can come to an answer and understand how we got there.

Part A:

Equation should be: f(x)=-17cos{(π/82)*(x-50))+28, where x is in cm

A(Amplitude): 17cm or 0.17m

VS(Vertical Shift/Midline): 28cm or 0.28m

P(Period): 1640cm or 16.4m

F(Frequency): 1/1640cm or 1/16.4m

PS(Phase Shift): 500 cm or 5m

DBCP(Distance Between Critical Points): 410cm or 4.1m

CV(Critical Values): 90+410n cm (5, 9.1, 13.2, 17.3, 21.4 are all good answers if they are in meters)

Part B:

f(0)=≈33.746cm or 0.33746m

(Be careful because the problem gives you two different units of measurement!(cm and m))

Now... How did we get all of that?

***For this problem I will be doing everything in centimeters.***

(A)The Amplitude is the distance between the vertical shift, or midline, of the sinusoidal function and a Minimum or Maximum of the function. You can get this value by taking the average of the Minimum and Maximum. In this problem, we got both of those from the problem, 11cm and 45cm respectively. Taking the average, we get (45-11)/2 or 17cm.

(VS)The Vertical Shift is the same at the Midline and is the centerline of the sinusoidal function. which we can get by adding the amplitude to the Minimum or Maximum of our sinusoidal function: 11+17=28 OR 45-17=28 (cm)

(P)The Period is how long it takes to complete 1 full cycle of the sinusoidal function. In this case, the problem also gives us this value at 16.4m or 1640cm.

(F)The Frequency is always the reciprocal of the Period: 1/1640cm.

(PS)The phase shift is how much the sinusoidal function is shifted left or right. For this problem, I believe it is easiest to try to model our scenario using a cosine function. This is because cosine has a maximum value when there is a value of 0 for our independent variable (In most cases we are talking about what is on the x-axis. In this case it is meters traveled.) We can see that we reach a minimum value after 5m or 500cm. If we move that high point forward to 500cm and flip it upside-down, we create our point we are after. We also know we should flip our cosine function upside-down because in terms of the problem, we are supposed to start by pushing our right foot downwards. Since we "shifted" our cosine function forward 500cm, we can call our Phase Shift 500cm.

(DBCP) The Distance Between Critical Points is found with knowing that there will be 4 critical points per period. If we divide our period by 4, we can determine this value: 1640cm/4=410cm

(CV)The Critical Values are found by either adding or subtracting the distance between them from any Minimum, Maximum, or Midpoint of our function. In this case, I will add 41cm to 50cm until I complete 1 period to get a few. 500cm, 910cm, 1320cm, 1730cm, 2140cm. Theoretically, we can get values like -320. However, we are not going in a negative direction so it would be weird to list this as a value. This is one example of why knowing the context of the problem will help our answers. But since we are traveling a distance(always positive) from the start of our movement, a more generic answer would be our smallest positive critical point plus our distance between the critical points times any natural constant, or 90+410n cm.

Creating our Equation:

NOTE: We will use a lot of these values above, so it is important to understand what they are and how to get them.

The general form of a sinusoidal equation goes a little something like this:

(+/-)Amplitude*Sinusoidal Function(Normal Period of the Sinusoidal Function/Period*(Independent Variable(+/-)Phase Shift)(+/-)Vertical Shift

The generic cosine function looks like this: f(x)=cos(x)

From this, we can see that the amplitude is 1, the period does not change (2π/2π=1 and is not written), and there is no shift of any kind.

We had decided earlier that is would be easiest to go with an upside-down cosine function, so we should start with something like this: f(x)=-cos(x) and go from there. Plugging in everything we know, we get something like this:

f(x)=-17*cos((2π/1640)*(x-500))+28 or f(x)=-17*cos((π/82)*(x-50))+28 when simplified.

So how high was your foot when you first started? Well you need to know where you started. Since we are talking about only traveling 500 centimeters, you must have started 500 centimeters before that, or at 0cm. Since we now have a function that will get the height of your foot at any distance traveled, we can plug 0cm into the equation and solve. As you may have guessed, it is not a pretty number.

If you are not permitted to use a calculator, just setting it up and simplifying should be sufficient: f(0)=-17*cos((π/82)*(-50))+28

Using a calculator, the answer comes out to be around 33.746cm.

I also want to mention that since the independent variable units cancel and the dependent variable units take over, doing this in half meters and half centimeters will not throw off your equation.

I hope this helps!

Answer 2

I believe part a is correct, except:

Cosine of what? Should this be cos(θ)?

Why didn't you draw the graph?

Part b.

One full revolution drives the bike forward 16.4 meters. A half revolution (pedaling from top to bottom) drives the bike forward half of that distance, or 8.2 meters.

Use f(x) = -17cos((π/820)*(x-90)) + 28. At f(0), y = 33.746 cm, approximately.

Here is a Desmos graph. The bottom of the first cycle is exactly at 500 cm.

https://www.desmos.com/calculator/5yl11swuy8