According to the manual of a certain car, a maximum torque of magnitude 76.0 N · m should be applied when tightening the lug nuts on the vehicle. If you use a wrench of length 0.276 m and you apply the force at the end of the wrench at an angle of 75.00° with respect to a line going from the lug nut through the end of the handle, what is the magnitude of the maximum force you can exert on the handle without exceeding the recommendation?

To solve this problem, we need to first understand the concept of torque and how it relates to force and distance.

Torque is the product of force and distance. It is a measure of the rotational force or moment of a given object. The formula for torque is:

Torque = Force × Distance × sin(θ),

where Force is the force applied, Distance is the distance from the rotating point (in this case, the lug nut) to the point where the force is applied (in this case, the end of the wrench), and θ is the angle between the line connecting the two points and the line of action of the force.

In this problem, we know the maximum torque (76.0 N · m), the length of the wrench (0.276 m), and the angle (75.00°). We need to find the maximum force that can be exerted.

Let's start by rearranging the formula to solve for the force:

Force = Torque / (Distance × sin(θ)).

Plugging in the known values:

Force = 76.0 N · m / (0.276 m × sin(75.00°)).

Now, we can calculate the force using a calculator:

Force ≈ 111.7 N.

Therefore, the magnitude of the maximum force that can be exerted on the handle without exceeding the recommendation is approximately 111.7 Newtons.