answer:Let's assume there were initially "n" girls, and the average weight of these girls was "A" kg. When the new girl comes in, the average weight increases by 2 kg, so the new average weight is "A + 2" kg. The total weight of the initial group of girls is n * A kg. The total weight of the group after the new girl joins is n * (A + 2) kg. The difference in total weight is due to the new girl replacing the girl who weighed 40 kg. So, the weight of the new girl minus the weight of the girl she replaced (40 kg) is equal to the increase in total weight: Weight of new girl - 40 kg = n * (A + 2) - n * A Given that the weight of the new girl might be 80 kg, we can substitute this value into the equation: 80 kg - 40 kg = n * (A + 2) - n * A 40 kg = n * 2 n = 40 kg / 2 n = 20 So, there were initially boxed{20} girls.

answer:From the given information, we know that the focus of the parabola y^{2}=4x is F(1,0). Since the line AB has a slant angle of 45^{circ}, its slope is 1, and thus its equation is y=x-1. By substituting the equation of line AB into the equation of the parabola, we can eliminate y to obtain: (x-1)^{2}=4x, which simplifies to x^{2}-6x+1=0. Let x_{1} and x_{2} be the roots of this quadratic equation, corresponding to the x-coordinates of the intersection points A(x_{1}, y_{1}) and B(x_{2}, y_{2}). Using Vieta's formulas, we have x_{1}+x_{2}=6 and x_{1}x_{2}=1. Now, we can calculate the length of line segment |AB| using the distance formula: |AB| = sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}} = sqrt{(x_{2}-x_{1})^{2}+(x_{2}-x_{1})^{2}} = sqrt{2(x_{2}-x_{1})^{2}} = sqrt{2}sqrt{(x_{2}-x_{1})^{2}} Recall that (x_{2}-x_{1})^{2} = (x_{2}+x_{1})^{2}-4x_{1}x_{2} = 36-4 = 32. Hence, |AB| = sqrt{2}sqrt{32} = 8. Therefore, the length of the line segment |AB| is boxed{8}.

answer:To solve this problem, we need to analyze some simple geometric properties of the ellipse, particularly focusing on its directrices. Given the standard equation for an ellipse dfrac{x^2}{a^2} + dfrac{y^2}{b^2} = 1, where a and b are the lengths of the semi-major and semi-minor axes, respectively, and c is the distance from the center to a focus. For an ellipse centered at the origin, the directrices are given by the equations x = pm dfrac{a^2}{c} for horizontal major axis and y = pm dfrac{a^2}{c} for vertical major axis. In our case, we have an ellipse with the equation dfrac{x^2}{8} + dfrac{y^2}{9} = 1. From this, we can deduce that a^2 = 9 and b^2 = 8. To find c, we use the relationship c^2 = a^2 - b^2, which leads to c^2 = 9 - 8 = 1 and thus c = 1. Now, since our ellipse has a vertical major axis (because b^2 < a^2), the directrices are defined by the equation y = pm dfrac{a^2}{c}. Substituting a^2 = 9 and c = 1, we get the equations of the directrices as y = pm 9. The distance between the two directrices is simply the difference of the y values, which is 9 - (-9) = 18. Therefore, the distance between the two directrices of the given ellipse is boxed{18}.

answer:Let's denote the number of coconuts Bruno can carry at a time as B. Since Barbie and Bruno make 12 trips together, and Barbie can carry 4 coconuts at a time, the total number of coconuts Barbie carries over the 12 trips is 12 trips * 4 coconuts/trip = 48 coconuts. The total number of coconuts carried by both Barbie and Bruno together in 12 trips is 144 coconuts (the entire pile). If we subtract the number of coconuts Barbie carries from the total, we get the number of coconuts Bruno carries: 144 coconuts (total) - 48 coconuts (carried by Barbie) = 96 coconuts (carried by Bruno). Now, we know that Bruno carries these 96 coconuts in 12 trips, so we can find out how many coconuts he carries per trip by dividing the total number of coconuts he carries by the number of trips: 96 coconuts / 12 trips = 8 coconuts/trip. Therefore, Bruno can carry boxed{8} coconuts at a time.