# An object with a mass of #7 kg# is pushed along a linear path with a kinetic friction coefficient of #u_k(x)= 4+secx #. How much work would it take to move the object over #x in [0, (pi)/12], where x is in meters?

The work is

We need

The frictional force is

The normal force is

So,

But,

The work done is

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To find the work done to move the object over the given interval, integrate the product of the force of kinetic friction (μ_k) and the distance moved (x) with respect to x over the interval [0, π/12]:

[ W = \int_{0}^{\frac{\pi}{12}} \mu_k(x) \cdot x , dx ]

First, find the expression for μ_k(x) by substituting the given function:

[ \mu_k(x) = 4 + \sec(x) ]

Now, integrate the expression over the given interval:

[ W = \int_{0}^{\frac{\pi}{12}} (4 + \sec(x)) \cdot x , dx ]

Once you integrate this expression, you'll get the work done to move the object over the given interval.

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