Last spring, I set out to ride my bicycle to the beach. Unfortunately, my front tire picked up a tack and I had to stop riding after another 15 minutes. Given that I rode at a constant speed of 6 mph and the radius of my bicycle tires is 2 feet, how high above the ground was the tack when I stopped.
To determine the height above the ground of the tack, we can visualize the situation as a circular path with a radius of 2 feet. When the tack was picked up by the front tire, it must have traveled a distance equivalent to 15 minutes of riding at 6 mph.
First, let's calculate the distance traveled in 15 minutes. Since we know that speed equals distance divided by time, we can rearrange the formula to solve for distance: distance = speed × time.
The speed is given as 6 mph, which we can convert to feet per minute: 6 mph * (5280 feet/1 mile) * (1 hour/60 minutes) = 528 feet/minute.
The time is given as 15 minutes.
Therefore, the distance traveled is: distance = 528 feet/minute * 15 minutes = 7920 feet.
Since the distance around a circle is equal to the circumference, we know that the distance traveled is equal to 7920 feet = 2πr, where r is the radius of the circle.
Rearranging the formula to solve for r, we get: r = distance / (2π) = 7920 feet / (2π) ≈ 1258.99 feet.
Since the radius of the bicycle tire is given as 2 feet, we can calculate the height of the tack above the ground as the difference between the radius of the circle formed by the tack's position and the radius of the bicycle tire: 1258.99 feet - 2 feet = 1256.99 feet.
Therefore, the tack was approximately 1256.99 feet above the ground when you had to stop riding.
To find the height of the tack above the ground, we can use the equations of motion for a rolling wheel.
First, let's find the time it took for the tack to pick up.
We know that speed (v) = 6 mph = 6 miles per hour = 6 * 5280 feet per 60 minutes = 528 feet per 60 minutes = 8.8 feet per minute.
We are given that the tack was picked up 15 minutes after the start, so the distance covered (d) = speed (v) * time (t).
Therefore, the distance covered = 8.8 feet per minute * 15 minutes = 132 feet.
Next, let's find the circumference (C) of the tire.
The circumference (C) of a circle = 2 * π * radius (r).
Given that the radius (r) = 2 feet, we have C = 2 * 3.14 * 2 = 12.56 feet.
Now, let's find the number of rotations (n) made by the tire.
The number of rotations (n) = distance covered (d) / circumference (C).
Therefore, the number of rotations = 132 feet / 12.56 feet = 10.51 rotations.
Since there are 2π radians in one revolution, we can find the total angle turned (θ) by multiplying the number of rotations (n) by 2π.
The total angle turned (θ) = 10.51 rotations * 2 * 3.14 radians per rotation = 65.99 radians.
Now, let's find the arc length (s) of the distance traveled along the circumference of the tire.
The arc length (s) = radius (r) * angle turned (θ).
Therefore, the arc length = 2 feet * 65.99 radians = 131.98 feet.
Lastly, let's find the height (h) of the tack above the ground.
The height (h) is the difference between the radius (r) and the arc length (s).
Therefore, the height = radius (r) - arc length (s) = 2 feet - 131.98 feet = -129.98 feet.
Since the calculated height is negative, it suggests that the tack was below the ground level. This indicates that there might be an error in the calculations or assumptions made. Please double-check the values and assumptions used in the problem.