i dontget how the textbook got the answer, h = -30cos[(1.43)x]° + 40
the question is:
a paintball is shot at a wheel of radius 40 cm. the paintball leaves a cricuarmark 10 cm from the outer edge of the wheel. As the wheel rolls, the mark moves in a circular motion.assuming that the paintball mark starts at its lowest point, determine the qeuation of the sinusoidal function that describes the hieght of the mark in terms of the distance the wheel travels.
11 years late but still, if anyone needs this in the future.
Assuming you already know about transformations from parent function:
f(x) = cosx
We have to figure out the max and min height of the mark.
Since the mark is 10cm away from the edge of the tire, its min height is 10cm above ground
Since the tire is actually 80cm high (40 is the radius, 80 is the diameter), the max height for the mark is 70 (10cm away from 80)
With the information, we know that the amplitude of the wave function is at 30cm, since it's half the distance between the max and min.
We know that the axis (equilibrium) is at 40cm, since it's equidistance from max and min.
Since the mark starts at minimum, the function is reflected in the x-axis (since cosx starts at max) making the "30" in the equation a negative.
To find the period, which is the length it takes for the tire to go back to the exact same position, we need to find the circumference. The circumference of tire is 2π*40, which is 80π. The circumference is in fact the length the tire will travel after one revolution.
Imagine if the tire is filled with paint around it, once the tire rolls around exactly one loop, its circumference smears the paint on the ground. The length of that paint is the length the tire has traveled, which is equal to its circumference.
Since period is 80π, in terms of degree, it is 360/(80π), this makes the 1.43. (If this is confusing, it's just the rules for transforming from cosx)
So now we have all the information to state the equation of the graph:
Amplitude = 30 (Negative as we start in the minimum)
Axis (equlibrium) = 40
Period = 360/(80π), which is about 1.43
Let h be the height of the mark (dependent variable)
Let x be the distant traveled by the tire (independent variable)
h = -30cos(1.43x) + 40
or more precisely
h = -30cos(180/(80π) * x) + 40
Your description is confusing.
You say the wheel rolls, but then the mark would trace out what is called a cycloid.
Your equation suggests that the mark rotates as the wheel rotates.
look at
h = -30cos(1.43x) + 40
no matter what we use for x, the max of cos(?) = 1
and the min of cos(?) = -1
so the max is -30(-1) + 40 = 70
and the min is -30(1) + 40 = 10
I don't see where the 1.43 comes from, was there a time given for one rotation?
I realize this is 4 years too late but it is 1.43 because you find the value of k in y=aCos(k(x-d) + c by calculating 360/period.
The period here is 80pi, which is the circumference of the circle. Thus after one full rotation of the wheel you get one cycle/period.
Therefore k = 360/80pi = 1.43
nope.... ust the radius and the 10 cm from the outer edge is given.
I think the fact that the answer made a conversion into degrees from radians made it a little more difficult to decipher.
Since the x-axis stands for the distance travelled, we can use
x=rθ, or
θ=x/r
converting to degrees,
θ=(x/r)*(180/π)
=(x/40)*(180/π)
=9x/(2π)
which was truncated to 1.43x
I would have left the answer as 9x/(2π) for the sake of the students.
To determine the equation of the sinusoidal function that describes the height of the mark in terms of the distance the wheel travels, we need to analyze the given information and understand what the equation represents.
1. We are told that the wheel has a radius of 40 cm. Let's call this distance "r."
2. The paintball leaves a circular mark 10 cm from the outer edge of the wheel. This means the distance of the mark from the center of the wheel can be calculated as "r - 10."
3. We are asked to determine the height of the mark. Since the paintball starts at the lowest point, the height can be measured vertically from the center of the wheel. Let's call this height "h."
4. The mark moves in a circular motion as the wheel rolls. So, as the wheel travels a certain distance, let's call it "x," the height of the mark changes.
With these observations, we can form a right triangle with the wheel's radius, the distance of the paintball mark from the center, and the height of the mark as the sides. The angle between the height and the radius can be represented as "θ."
Now, let's consider the trigonometric function that relates the sides of a right triangle: "cosine." The cosine of an angle is the ratio of the adjacent side to the hypotenuse.
In this case, we have:
Cos(θ) = (r - 10) / r
To isolate the height "h," we can rearrange the equation as:
Cos(θ) = (r - 10) / r
Cos(θ) = 1 - 10 / r
Multiplying both sides by "r" to eliminate the fraction:
r * Cos(θ) = r - 10
Next, solving for "h," we need to consider that the height can be negative depending on the orientation. Hence, we have:
h = r - r * Cos(θ)
Substituting "Cos(θ)" with its corresponding trigonometric value:
h = r - r * Cos[(1.43) * x]°
Simplifying further:
h = r - r * Cos[(1.43) * x]
Therefore, the equation that describes the height of the mark in terms of the distance the wheel travels is:
h = -30 * Cos[(1.43) * x]° + 40
This equation shows the relationship between the distance the wheel travels (x) and the height of the mark (h). The cosine function gives the varying height as the wheel rotates.