answer:If there are 150 cows in each herd and there are 32 herds, then the total number of cows can be calculated by multiplying the number of cows in one herd by the number of herds. Total number of cows = Number of cows in one herd × Number of herds Total number of cows = 150 × 32 Total number of cows = 4800 So, they have a total of boxed{4800} cows on the farm.

answer:To solve ( f(3) ), substitute ( x = 3 ) into the polynomial: [ f(3) = 9(3)^4 + 7(3)^3 - 5(3)^2 + 3(3) - 6 ] Simplify each term: [ 9(3)^4 = 9 cdot 81 = 729 ] [ 7(3)^3 = 7 cdot 27 = 189 ] [ -5(3)^2 = -5 cdot 9 = -45 ] [ 3(3) = 9 ] Add these results together: [ f(3) = 729 + 189 - 45 + 9 - 6 = 876 ] Thus, ( f(3) = boxed{876} ).

answer:Since f(x) is an odd function, we have f(-x) = -f(x). Given the functional equation f(4 + x) + f(-x) = 0, we can write: f(4 + x) = -f(-x) = f(x). This implies that the period of the function is 4, meaning f(x + 4k) = f(x) for any integer k. Now, let's find f(2011), f(2012), and f(2013) using the periodic property: begin{align*} f(2011) &= f(503 times 4 - 1) = f(-1) = -f(1) text{ (due to odd symmetry)}, f(2012) &= f(503 times 4 + 0) = f(0) text{ (since any multiple of 4 is a period)}, f(2013) &= f(503 times 4 + 1) = f(1). end{align*} Now we can sum these up: begin{align*} f(2011) + f(2012) + f(2013) &= -f(1) + f(0) + f(1) &= -9 + f(0) + 9 text{ (since it's given that } f(1) = 9). end{align*} Because f(x) is an odd function, we have that f(0) = 0. Therefore: f(2011) + f(2012) + f(2013) = -9 + 0 + 9 = boxed{0}. Hence, the correct answer is D.

answer:Since sin(-80^circ) = k, by using the property of sine function which is odd, we have sin(80^circ) = -k. Now, we need to find cos(80^circ). For any angle theta, the relationship between sine and cosine is given by the Pythagorean identity sin^2(theta) + cos^2(theta) = 1. Therefore, cos^2(80^circ) = 1 - sin^2(80^circ) = 1 - (-k)^2 = 1 - k^2. Taking the square root and considering that cos(80^circ) is positive in the first quadrant (as 80^circ lies in the first quadrant), we get cos(80^circ) = sqrt{1 - k^2}. To find tan(100^circ), we use the angle sum identity for tangent: tan(100^circ) = tan(180^circ - 80^circ) = -tan(80^circ) = -frac{sin(80^circ)}{cos(80^circ)} = -frac{-k}{sqrt{1-k^2}} = boxed{frac{k}{sqrt{1-k^2}}}. The correct choice is C.