A Person is riding a bike with wheels that have a 14 inch radius at 15 mph. The bicyclist rides over some gum which sticks in the tire, assume this is t=0.
1. Find a function that gives the height in inches above the ground of the gum, with input is second
To find the function that gives the height in inches above the ground of the gum over time, we need to consider the motion of the bike and the rotation of the wheels.
First, let's find the circumference of the tire. The circumference of a circle is given by the formula: C = 2πr, where r is the radius of the circle.
In this case, the radius of the wheel is given as 14 inches, so the circumference is:
C = 2π(14) = 28π inches
Since the bike is traveling at a speed of 15 mph, we need to convert this to inches per second to match the time input for the function. There are 60 minutes in an hour and 60 seconds in a minute, so:
15 mph = 15 * 5280 feet per hour = 15 * 5280 * 12 inches per hour
= 15 * 5280 * 12 / 60 / 60 inches per second
≈ 352 inches per second
Now, let's consider the rotation of the wheel. As the wheel makes a complete revolution, the gum will return to its original position (height above the ground). Therefore, one complete revolution of the wheel corresponds to one period of the function.
Since the circumference is 28π inches, the height of the gum will change by this amount as the wheel completes one revolution.
Now, to find the height of the gum as a function of time, we divide the distance the bike has traveled by the circumference of the tire:
h(t) = (distance traveled by bike) / (Circumference of the tire)
Given that the bike is traveling at a constant speed of 352 inches per second, the distance traveled after time t is:
distance traveled = speed * time = 352t
Plugging this into the function, we have:
h(t) = 352t / (28π)
Therefore, the function that gives the height in inches above the ground of the gum, with input in seconds, is h(t) = 352t / (28π).
To find a function that gives the height in inches above the ground of the gum at a given time, let's break down the problem step-by-step:
Step 1: Determine the circumference of the bike tire.
The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle (in this case, 14 inches). Thus, the circumference of the bike tire is:
C = 2π(14) = 28π inches.
Step 2: Calculate the period of the bike's motion.
The period refers to the time it takes for the bike to complete one full revolution. In this case, the bike is traveling at 15 mph, so the period can be calculated as:
Period = distance / speed = C / speed = (28π) / 15.
Step 3: Convert the period into seconds.
Since we want to find the height in inches above the ground for each second, we need to convert the period to seconds. We know that 1 hour = 60 minutes and 1 minute = 60 seconds, so:
Period (in seconds) = Period (in hours) × 60 × 60 = Period / 15 × 60 × 60.
Step 4: Create the function.
Now that we have the period in seconds, we can create the function. Let's call h(t) the height in inches above the ground of the gum at time t (in seconds). Since the bike's tire completes one full revolution in the period, the height function can be defined as:
h(t) = (2πr) × (t / period) = 28π × (t / Period).
So, the function that gives the height in inches above the ground of the gum as an input of time t (in seconds) is:
h(t) = 28π × (t / Period).
Replace "Period" in the formula with the value calculated in Step 3.