# How much work would it take to push a # 1 kg # weight up a # 3m # plane that is at an incline of # pi / 4 #?

This requires 20.8 J of work done.

The definition of work is

Here, the force we work against is the force of gravity, so the angle in the above formula is the one between the incline and the vertical.

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To find the work done in pushing the weight up the incline, you can use the formula:

[ W = F \times d \times \cos(\theta) ]

Where:

- ( W ) is the work done (in joules)
- ( F ) is the force applied parallel to the incline (in newtons)
- ( d ) is the displacement along the incline (in meters)
- ( \theta ) is the angle of the incline (in radians)

Given:

- ( m = 1 ) kg (mass of the weight)
- ( d = 3 ) m (displacement along the incline)
- ( \theta = \frac{\pi}{4} ) radians

To find ( F ), you can resolve the weight force (( mg )) into two components: one parallel to the incline (( mg \sin(\theta) )) and one perpendicular (( mg \cos(\theta) )). Since the weight is being pushed up the incline, the force opposing the motion is the parallel component (( mg \sin(\theta) )). Thus:

[ F = mg \sin(\theta) ]

Substituting the given values:

[ F = (1 , \text{kg}) \times (9.81 , \text{m/s}^2) \times \sin\left(\frac{\pi}{4}\right) ]

[ F \approx 9.81 , \text{N} \times 0.707 ]

[ F \approx 6.94 , \text{N} ]

Now, plug the values into the work formula:

[ W = (6.94 , \text{N}) \times (3 , \text{m}) \times \cos\left(\frac{\pi}{4}\right) ]

[ W \approx 6.94 , \text{N} \times 3 , \text{m} \times 0.707 ]

[ W \approx 14.71 , \text{J} ]

So, it would take approximately 14.71 joules of work to push the 1 kg weight up a 3 m plane at an incline of ( \frac{\pi}{4} ) radians.

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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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