# A truck pulls boxes up an incline plane. The truck can exert a maximum force of #4,500 N#. If the plane's incline is #pi/8 # and the coefficient of friction is #4/3 #, what is the maximum mass that can be pulled up at one time?

The mass is

Resolving in the direction parallel to the plane

Let the force exerted by the truck be

Let the frictional force be

The coefficient of friction

The normal force is

The angle of the plane is

The acceleration due to gravity is

Therefore,

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To find the maximum mass that can be pulled up the incline, we need to consider the forces acting on the box. The maximum force the truck can exert, 4500 N, is opposed by the force of friction and the component of the weight of the box acting down the incline.

The force of friction can be calculated using the equation:

( F_{friction} = \mu \cdot F_{normal} )

Where ( \mu ) is the coefficient of friction and ( F_{normal} ) is the normal force. The normal force can be found using trigonometry:

( F_{normal} = mg \cdot cos(\theta) )

Where ( m ) is the mass of the box, ( g ) is the acceleration due to gravity (9.8 m/s²), and ( \theta ) is the angle of the incline (π/8).

The component of the weight of the box acting down the incline can be found using:

( F_{down} = mg \cdot sin(\theta) )

At maximum capacity, the force the truck exerts (4500 N) equals the sum of the force of friction and the component of the weight of the box down the incline:

( F_{truck} = F_{friction} + F_{down} )

Substituting the equations for friction and the downward force, and solving for mass:

( 4500 N = \mu \cdot F_{normal} + mg \cdot sin(\theta) )

( 4500 N = \mu \cdot (mg \cdot cos(\theta)) + mg \cdot sin(\theta) )

( 4500 N = \frac{4}{3} \cdot (mg \cdot cos(\frac{\pi}{8})) + mg \cdot sin(\frac{\pi}{8}) )

( m = \frac{4500 N}{(\frac{4}{3} \cdot (g \cdot cos(\frac{\pi}{8})) + g \cdot sin(\frac{\pi}{8}))} )

Calculating ( m ) gives us the maximum mass that can be pulled up the incline at one time.

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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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