To model the height of the nail from the ground using the cosine function, we can assume that the ground level corresponds to the x-axis, and the positive y-axis represents the upward direction. Let's denote the height of the nail as "h" and the distance traveled by Carlos as "x" (measured horizontally along the ground).
Given:
- Diameter of the tire = 2.5 feet
Using the cosine function, we can relate the angle of rotation of the tire (θ) to the height of the nail (h). The angle θ can be determined based on the distance traveled by Carlos (x) and the radius of the tire (r), which is half the diameter.
Since the diameter is 2.5 feet, the radius (r) is 2.5 / 2 = 1.25 feet.
Using the cosine function, we have:
cos(θ) = adjacent side / hypotenuse
In this case, the adjacent side represents the height of the nail (h), and the hypotenuse corresponds to the radius of the tire (r).
Therefore, we can write the equation:
cos(θ) = h / r
Substituting the values:
cos(θ) = h / 1.25
Simplifying the equation:
h = 1.25 * cos(θ)
This equation models the height of the nail (h) from the ground in terms of the cosine of the angle of rotation (θ) of the tire.
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