Question

A humming bird starts 10 meters away from a feeder at time zero. Moving at a constant rate, the humming bird takes 5 seconds to reach the feeder. It remains at the feeder for 4 seconds and then moves away from the feeder at a constant rate. One second after it leaves the feeder, it is 5 meters from the feeder. Total time elapsed is 10 seconds

A) Write a piece-wise function that describes the humming birds distance from the feeder as a function of time (d(t)= ?).

B) Graph your function.

C) Determine the domain of d(t).

Hoo-Boy I haven't done one of these in awhile...

Answer 1

Even if you can help me with part A that's great, I can probably do B and C if I have that... I just don't know how to do it.

Answer 2

There are three time intervals here.
Can you tell what they are?

Answer 3

5 seconds, 4 seconds, and 1 second, right?

Answer 4

Good, but 5 seconds and 4 seconds are those compete intervals.
After the 4-second interval, it starts doing something in which at 1 sec we know the position.
Let's start with the 5-sec interval.

for \(0 \le t \le 5\)

Answer 5

It starts 10 m away from the feeder and reaches the feeder in 5 sec.
What speed is it traveling at?

Answer 6

Erm... Shoot I'm honestly not sure ^^"

Answer 7

Remember what speed is:

\(speed = \dfrac{distance}{time} \)

Answer 8

OH okay, hang on let me do that math...

Answer 9

So it's traveling at a speed of 2 seconds, yes?

Answer 10

A speed is a distance divided by a time, so the units are the units of distance, or length, divided by units of time.

\(speed = \dfrac{distance}{time} = \dfrac{10~m}{5~sec} = 2 \dfrac{m}{sec} \)

In the first interval of time, from 0 seconds to 5 seconds,
it is traveling at a speed of 2 m/sec

Answer 11

2 seconds is not a speed. It is an amount of time or an interval of time.

Answer 12

Ok. Now we need an equation for the distance from the feeder or the interval of 0 to 5 sec.

Answer 13

Since we have a constant rate, the speed being constant, we need an equation that has a constant rate of change, or slope. A linear equation, whose graph is a straight line, has a constant slope.

Here is the equation of a line in the slope-intecept form:

\(y = mx + b\)

where m = slope, and b = y-intercept

Answer 14

When the time is 0, the bird is at 10 m.

\(y = mx + b\)

\(10 = m(0) + b\)

\(10 = b\)

Since we get b = 0, we can use that in our equation.
Our equation will have the form

\(y = mx + 10\)

We need the slope.
As I mentioned above, the slope is the constant rate, which in this case the constant speed.

As the bird flies in the first interval of time, it gets closer tot eh feeder, so the distance becomes smaller, until at 5 seconds it is at the feeder and the distance is zero.

That means we need to use a -2 for the slope.

We get

\(y = -2x + 10\)

Let x = 0, and see what the distance is.
Then let x = 5, and see what the distance is.

Answer 15

y = 10 and y = 0 if we plug in x with 0 and then 5

Answer 16

\(y = -2x + 10\)

For x = 0,

\(y = -2(0) + 10\)

\(y = 0 + 10\)

\(y = 10\)

This confirms that at time 0 sec, the distance is 10 m.

For x = 5,

\(y = -2(5) + 10\)

\(y = -10 + 10\)

\(y = 0\)

This confirms that at time 5 sec, the distance is 0 m, and the bird is at the feeder.

Answer 17

You are correct.
That means we have the equation for the first interval.

Answer 18

Remember the problem uses d(t) instead of y, and t instead of x, so let's write it the way the problem asks.