Problem

Barry and Timmy are using a ramp to load furniture into a truck. While rolling a 270-pound piano up the ramp, they discover that the truck is too full with other furniture for the piano to fit. Timmy holds the piano in place on the ramp while Barry repositions other items to make room for it in the truck. If the angle of inclination of the ramp is $20^{\circ}$, how many pounds of force must Timmy exert to hold the piano in position? Timmy must exert pounds of force. (Type an integer or decimal rounded to one decimal place as needed.)

Solution

Step 1 :Given that the weight of the piano is 270 pounds and the angle of inclination of the ramp is 20 degrees.

Step 2 :Convert the angle from degrees to radians. This is done because the trigonometric functions in Python use radians, not degrees. The conversion is done by multiplying the angle by \(\frac{\pi}{180}\). So, the angle in radians is approximately 0.349 radians.

Step 3 :The force required to hold the piano in place on the ramp is equal to the weight of the piano times the sine of the angle of inclination of the ramp. This is because the force required to hold the piano in place is equal to the component of the weight of the piano that is acting parallel to the ramp, which is given by the weight times the sine of the angle of inclination.

Step 4 :Calculate the force using the formula: \(Force = weight \times \sin(angle)\). Substituting the given values, we get the force as approximately 92.3 pounds.

Step 5 :So, the force that Timmy must exert to hold the piano in position is approximately \(\boxed{92.3}\) pounds.

From Solvely APP
Source: https://solvelyapp.com/problems/24046/

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