answer:Using the given functional equation and setting ( x = 50 ) and ( y = 1.5 ), we get [ g(75) = frac{g(50)}{(1.5)^2}. ] Substituting the value of ( g(50) = 25 ), [ g(75) = frac{25}{(1.5)^2} = frac{25}{2.25} = frac{100}{9}. ] Thus, the value of ( g(75) ) is ( boxed{frac{100}{9}} ).

answer:First, convert the point P(2, frac{pi}{3}) to rectangular coordinates, which gives P(1, sqrt{3}). Next, convert the polar equation of the line to rectangular form. The line rho(cos theta + sqrt{3} sin theta) = 6 converts to x + sqrt{3}y - 6 = 0. Now, we can use the formula for the distance d between a point (x_1, y_1) and a line Ax + By + C = 0, which is given by d = frac{|Ax_1 + By_1 + C|}{sqrt{A^2 + B^2}}. Substituting our values, we get d = frac{|1 + 3 - 6|}{sqrt{1 + (sqrt{3})^2}} = 1. Therefore, the distance from the point to the line is boxed{1}. This problem tests the ability to convert between polar and rectangular coordinates, as well as the application of the distance formula for a point to a line. It requires both reasoning and computational skills and is of moderate difficulty.

answer:To calculate the total number of fruits remaining in all the trees after Dulce picks 2/5 of the oranges from each tree, we proceed as follows: 1. **Calculate the total number of oranges before picking:** The total number of oranges in all the trees before Dulce picked any is calculated by multiplying the number of trees by the number of oranges per tree. This gives us: [ 8 , text{trees} times 200 , text{oranges/tree} = 1600 , text{oranges}. ] 2. **Calculate the number of oranges Dulce picks from each tree:** Since Dulce picks 2/5 of the oranges from each tree, and each tree has 200 oranges, the number of oranges she picks from each tree is: [ frac{2}{5} times 200 , text{oranges} = 80 , text{oranges}. ] 3. **Calculate the total number of oranges Dulce picks from all trees:** Since there are 8 trees, and she picks 80 oranges from each tree, the total number of oranges she picks from all the trees is: [ 8 , text{trees} times 80 , text{oranges/tree} = 640 , text{oranges}. ] 4. **Calculate the total number of oranges remaining:** After picking 640 oranges from the trees, the total number of oranges remaining is calculated by subtracting the number of oranges picked from the total number of oranges before picking. This gives us: [ 1600 , text{oranges} - 640 , text{oranges} = 960 , text{oranges}. ] Therefore, the total number of fruits remaining in all the trees is boxed{960} oranges.

answer:Since the function f(x)=ax^{2}+bx+3a+b is an even function with the domain [a-1, 2a], we know that: 1. Even functions do not contain odd degree terms, which implies that b=0. 2. The domain of an even function is symmetrical about the origin. Equating the endpoints of the given domain, we get a-1=-2a, which yields a=dfrac{1}{3}. Now, substituting the values of a and b into the expression (a+b), we get: (dfrac{1}{3}+0) = boxed{dfrac{1}{3}}. Therefore, the answer is option D.