The coefficient of static friction between a 3.00kg crate and a ramp at an incline of 35.0degree is .30. what is the magnitude of the minimum force ,f, that must be applied to the crate parallel to the ramp to prevent the crate from sliding down the ramp

The minimun force is 9.65 [N]

Explanation:

The minimun force is the one that doesn't allow the crate sliding down the ramp.

First we need to draw the free body diagram with all the forces that are acting over the crate.

Then we set the directions in which the forces are possitives or negatives, and after that, we can develop the equations based on the movements in the Y-axis and the X-axis

At this point we will have one equation with the unknwon value Fmin. And we can resolve this unknown value with the deffinition of fiction force.

We replace this new value in the original expression, and we can determinate the Fmin.

For a better undertanding, in the attached images you can find all the solving process.