# A truck pulls boxes up an incline plane. The truck can exert a maximum force of #3,500 N#. If the plane's incline is #(5 pi )/12 # and the coefficient of friction is #5/12 #, what is the maximum mass that can be pulled up at one time?

We're asked to find the maximum mass that the truck is able to pull up the ramp, given the truck's maximum force, the ramp's angle of inclination, and the coefficient of friction.

The forces acting on the boxes are

the truck's upward pulling force

We thus have our net force equation

Or

Therefore,

The problem gives us

Plugging in these values:

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To find the maximum mass that can be pulled up the incline, we first need to calculate the net force acting on the mass. The force exerted by the truck can be decomposed into two components: one parallel to the incline (the force pulling the mass up) and one perpendicular to the incline (the force pressing the mass against the incline). The force pulling the mass up the incline is given by the component of the truck's force parallel to the incline, which is F_parallel = F_truck * sin(theta), where theta is the angle of the incline. The frictional force resisting motion up the incline is given by F_friction = coefficient_of_friction * normal_force, where the normal force is the component of the weight of the mass perpendicular to the incline, which is equal to the weight of the mass times cos(theta). The net force pulling the mass up the incline is then the difference between the force pulling the mass up and the frictional force resisting motion. Finally, we use Newton's second law, F = m * a, where F is the net force, m is the mass, and a is the acceleration up the incline. Since the mass is being pulled up the incline, the acceleration is along the incline and has the same direction as the force pulling the mass up, so we can use the magnitude of the force pulling the mass up as the net force in the equation. Rearranging the equation to solve for the mass, we get m = F_net / g, where g is the acceleration due to gravity. Plugging in the given values, we have m = (F_truck * sin(theta) - coefficient_of_friction * m * g) / g. Solving for m gives us the maximum mass that can be pulled up the incline.

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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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