# An object is at rest at #(2 ,1 ,6 )# and constantly accelerates at a rate of #1/4 m/s^2# as it moves to point B. If point B is at #(3 ,2 ,7 )#, how long will it take for the object to reach point B? Assume that all coordinates are in meters.

In a 3-dimensional coordinate system, the distance between two points is given by:

We can use the equation of motion:

Note that the term 'r' I was using in the distance calculation above is the same as the distance 's' that the object moves. I don't want the symbol change to be confusing.

(I used 0.25 rather than 1/4 for ease of writing, but of course it means the same thing)

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To find the time it takes for the object to reach point B, we can use the kinematic equation:

[s = ut + \frac{1}{2}at^2]

Where: (s) = displacement, (u) = initial velocity (since the object is at rest, (u = 0)), (a) = acceleration, and (t) = time.

Given that (s) = displacement from (2, 1, 6) to (3, 2, 7), we can calculate the displacement using the distance formula:

[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}]

Substituting the coordinates, we find that (d ≈ \sqrt{(1^2 + 1^2 + 1^2)} ≈ \sqrt{3}).

Now, we plug the values into the kinematic equation:

[\sqrt{3} = 0 \cdot t + \frac{1}{2} \cdot \frac{1}{4} \cdot t^2]

Solving for (t):

[t^2 = 2 \cdot \sqrt{3}]

[t ≈ \sqrt{2 \cdot \sqrt{3}} ≈ 1.732] seconds.

Therefore, it will take approximately 1.732 seconds for the object to reach point B.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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