Questions: x(t) = 2 + 3/t, y(t) = t-1, 2 ≤ t ≤ 6

Transcript text: $x(t)=2+\frac{3}{t}, \quad y(t)=t-1,2 \leq t \leq 6$

Solution

Solution Steps

To solve the given problem, we need to evaluate the functions $ x(t) $ and $ y(t) $ over the interval $ 2 \leq t \leq 6 $. We can create a list of $ t $ values within this range and then compute the corresponding $ x(t) $ and $ y(t) $ values.

Step 1: Define the Functions

We have the functions defined as: \[ x(t) = 2 + \frac{3}{t} \] \[ y(t) = t - 1 \] for the interval $ 2 \leq t \leq 6 $.

Step 2: Evaluate the Functions

We evaluate $ x(t) $ and $ y(t) $ at 100 evenly spaced points between $ t = 2 $ and $ t = 6 $.

Step 3: Results

The computed values for $ t $, $ x(t) $, and $ y(t) $ are as follows:

For $ t = 2 $: \[ x(2) = 3.5, \quad y(2) = 1 \]
For $ t = 3 $: \[ x(3) \approx 3.0, \quad y(3) = 2 \]
For $ t = 4 $: \[ x(4) = 2.75, \quad y(4) = 3 \]
For $ t = 5 $: \[ x(5) = 2.6, \quad y(5) = 4 \]
For $ t = 6 $: \[ x(6) = 2.5, \quad y(6) = 5 \]

Final Answer

The values of $ x(t) $ and $ y(t) $ at selected points are:

At $ t = 2 $: $ x(2) = 3.5 $, $ y(2) = 1 $
At $ t = 3 $: $ x(3) \approx 3.0 $, $ y(3) = 2 $
At $ t = 4 $: $ x(4) = 2.75 $, $ y(4) = 3 $

Thus, the final boxed answers are: \[ \boxed{x(2) = 3.5, \quad y(2) = 1} \] \[ \boxed{x(3) \approx 3.0, \quad y(3) = 2} \] \[ \boxed{x(4) = 2.75, \quad y(4) = 3} \]

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