A fly starts out 2 meters from a light bulb, flies closer to the light then farther away, 2 meters again. At this point the fly goes toward the bulb again but hits the bulb and flies away. Find a function (it may be piecewise) that gives the distance of the bug from the light as a function of time.
That makes it pretty easy
Let y = distance from light bulb
y = - 1.5x + 2 ... for x = 0 to 1
y = 1.5x - 1 ... for x = 1 to 2
y = -2x + 6 ... for x = 2 to 3
y = 2x - 6 ... for x > 3
Well, this situation seems like a real "buggy" situation, doesn't it? But fear not, I'll try to shed some light on it for you (pun intended)!
Let's break down the journey of our little flying friend, the bug. We know that it starts 2 meters away from the light bulb. As it approaches the light, it goes closer and closer, until it reaches a minimum distance. Then, it goes away from the light again, reaches the same 2-meter distance, but unfortunately, this time it hits the bulb and flies away.
So, let's divide this into two phases:
Phase 1: Approaching the Light Bulb
During this phase, the bug is getting closer to the light bulb. We know it starts 2 meters away from the bulb, and let's say it takes t₁ seconds to reach the minimum distance. A simple linear function could represent this as:
f₁(t) = 2 - t/t₁
Phase 2: Moving Away from the Light Bulb
After hitting the bulb, our bug decides it had enough and starts to fly away. Here, we know the bug reaches a maximum distance of 2 meters away from the bulb again. Let's suppose it takes t₂ seconds to do so. Another linear function could describe this movement:
f₂(t) = 2 + (t-t₁)/t₂
Now, all we need to do is combine these two functions to get the overall distance function:
f(t) = { f₁(t) ; 0 ≤ t ≤ t₁
{ f₂(t) ; t₁ ≤ t
Voila! We've got ourselves a piecewise function that describes the distance of the bug from the light bulb as a function of time. Now that the "bug data" is illuminated for you, I hope it brings you some laughter and enlightenment!
Let's break down the problem into different intervals:
Interval 1: The fly starts 2 meters away from the light bulb and flies closer to the light.
Interval 2: The fly flies farther away from the light bulb, reaching a distance of 4 meters.
Interval 3: The fly turns around and starts flying back toward the light, but hits the bulb and flies away.
In each interval, we can consider the distance of the fly from the light bulb as a function of time.
Interval 1: The fly starts at a distance of 2 meters and flies towards the light bulb. Let's assume the fly flies at a constant speed of 1 meter per second. So, the distance of the fly from the light bulb in this interval can be represented by:
d(t) = 2 - t
Interval 2: The fly flies farther away from the light bulb, reaching a distance of 4 meters. Again, assuming the fly flies at a constant speed of 1 meter per second, the distance of the fly from the light bulb in this interval can be represented by:
d(t) = 2 + t
Interval 3: The fly turns around and starts flying back toward the light bulb. However, in this interval, the fly hits the bulb and flies away. To represent this interval, we can use a piecewise function. Let's assume the fly reaches the light bulb at time t = T, and the fly immediately flies away after hitting the bulb. So, the distance of the fly from the light bulb in this interval can be represented by:
d(t) = 0 for t < T
d(t) = t for t ≥ T
Therefore, the overall function that gives the distance of the fly from the light as a function of time is:
d(t) = 2 - t for 0 ≤ t < 2
d(t) = 2 + t for 2 ≤ t < 4
d(t) = 0 for t < T and t ≥ 4
d(t) = t for t ≥ T and t < 4
Please note that the value of T, which represents the time at which the fly hits the bulb, is not provided in the problem and would need to be determined based on additional information.
To find a function that gives the distance of the fly from the light as a function of time, let's break down the problem into different parts.
1. Fly flies closer to the light: Let's assume that the fly starts flying towards the light bulb at time t = 0. The initial distance from the light bulb is 2 meters, and it moves towards the light. We can define this part of the fly's motion as:
d(t) = 2 - vt
Here, d(t) represents the distance of the fly from the light bulb at time t, and v represents the speed of the fly towards the light bulb.
2. Fly flies farther away: At some point, the fly reaches the light bulb and starts flying away. Let's assume this happens when the distance is at a minimum (closest to the light bulb). We can represent the distance as:
d(t) = 2 + ut
Here, d(t) represents the distance of the fly from the light bulb at time t, and u represents the speed of the fly moving away from the light bulb.
3. Fly hits the bulb and flies away: At a certain time t = T, the fly hits the light bulb and starts moving away. From this point on, we can assume the fly is moving away from the light bulb at a constant speed. We can represent this as:
d(t) = 0 + wt
Here, d(t) represents the distance of the fly from the light bulb at time t, and w represents the speed of the fly flying away from the light bulb.
To summarize, the function that gives the distance of the bug from the light bulb as a function of time is:
d(t) =
2 - vt, if 0 ≤ t < T
2 + ut, if T ≤ t < T + Δt
0 + wt, if t ≥ T + Δt
Note that T represents the time at which the fly hits the light bulb, and Δt represents the duration of time the fly remains in contact with the bulb. The values of v, u, and w would need to be determined based on specific information about the fly's speed.