An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle θ with the plane, then the magnitude of the force is F=μW/μ sin θ + cos θ where μ is a positive constant called the coefficient of friction and where 0 ≤ θ ≤ π/2. Show that F is minimized when tan θ = μ.

What is the slope simplify your answer and write it as a proper fraction, improper fraction, or integer

Mrs. julien’s and mrs. castillejo’s classes are selling cookie dough for a school fundraiser. customers can buy packages of macadamia nut chip cookie dough and packages of triple chocolate cookie dough. mrs. julien’s class sold 25 packages of macadamia nut chip cookie dough and 30 packages of triple chocolate cookie dough for a total of $221.25. mrs. castillejo’s class sold 5 packages of macadamia nut chip cookie dough and 45 packages of triple chocolate cookie dough for a total of $191.25. (a) write the system of equations that model the problem. be sure to explain which equation represents which situation. (b) find the cost of each type of cookie. show your work. (c) explain which method you used to solve the system and why you chose that method.

Aregular hexagonal pyramid has a base area of 45 in2 and a lateral area of 135in 2. what is the surface are are of the regular hexagonal pyramid