answer:Solution: If y= frac {1}{3}x^{3}+x, then y'|_{x=1}=2, which means the equation of the tangent line to the curve y= frac {1}{3}x^{3}+x at the point (1, frac {4}{3}) is y- frac {4}{3}=2(x-1). The intersection points of this line with the coordinate axes are ( frac {1}{3},0) and (0,- frac {2}{3}). The area of the triangle formed by these points is frac {1}{9}. Therefore, the correct answer is boxed{A}. 1. First, use the geometric meaning of the derivative to find the slope of the tangent line at the point P(x_{0},y_{0}) on the curve, and then derive the equation of the tangent line. 2. Use the equation of the tangent line and the equations of the coordinate axes to find the coordinates of the intersection points. 3. Use the formula for the area to calculate the area. The geometric meaning of the derivative of the function y=f(x) at x=x_{0} is the slope of the tangent line to the curve y=f(x) at the point P(x_{0},y_{0}). The equation of the tangent line passing through point P is: y-y_{0}=f′(x_{0})(x-x_{0}).

answer:1. **Understanding the problem**: - We know there are 35 people who live above someone. - We know there are 45 people who live below someone. - The first floor has one third of all the residents in the house. 2. **Interpreting the conditions**: - People living above someone must be on the 1st or 2nd floor. - People living below someone must be on the ground or 1st floor. 3. **Analyzing counts**: - Let's denote: - (G) for the number of people on the ground floor, - (F_1) for the number of people on the 1st floor, - (F_2) for the number of people on the 2nd floor. 4. **Sum of people living above and below**: - The people living above are on the 1st and 2nd floors, hence (F_1 + F_2 = 35). - The people living below are on the ground floor and 1st floor, hence (G + F_1 = 45). 5. **Total count inclusion**: - The total count of residents in the house is (G + F_1 + F_2), but we notice (F_1) got counted in both the 35 and the 45 totals. - Summing these calculations, [ 35 + 45 = G + 2F_1 + F_2, ] we have counted the 1st floor residents twice. 6. **Total residents calculation**: - We need the net total summing: [ 35 + 45 = 80. ] - Since (F_1) is counted twice, the total population of the building: [ text{Total} = 80 - F_1. ] 7. **Expressing total in terms of (F_1)**: - By the given condition: ( F_1 text{ (first floor residents)} = frac{1}{3} text{ of total residents} ), [ F_1 = frac{1}{3}(80 - F_1). ] 8. **Solving the equation**: [ 3F_1 = 80 - F_1, ] [ 4F_1 = 80, ] [ F_1 = 20. ] 9. **Substituting back for the total number of residents**: - Using (G + F_1 + F_2 = 3F_1,) (text{Total residents} = 80 - 20 = 60). 10. **Verifying the solution**: - The given count of residents on the second floor ((G = 25)), - (G = 45 - 20 = 25), - (d = 35 - 20 = 15). Conclusion: The total number of people living in the house is ( boxed{60} ).

answer:To find an integer smaller than sqrt{23}, we first identify perfect squares near 23: - We know that 4^2 = 16 and 5^2 = 25. Using these perfect squares, we can establish the inequality 16 < 23 < 25. This implies that 4^2 < 23 < 5^2. Taking the square root of all parts of this inequality gives us 4 < sqrt{23} < 5. Therefore, the integers smaller than sqrt{23} include 4, but since the question asks for an integer smaller than sqrt{23}, we can choose 4 as our answer. However, it's important to note that the answer is not unique because any integer less than 4 also satisfies the condition. Thus, the answer is: boxed{4 text{ (answer not unique)}}.

answer:Since the graph of the function y = 4tan x intersects the graph of the function y = 6sin x at point P within the interval (0, frac{pi}{2}), we have that 4tan x = 6sin x. This can be rewritten using the sine and cosine relationship tan x = frac{sin x}{cos x} as follows: 4 cdot frac{sin x}{cos x} = 6sin x. Simplifying this equation, we get: 4 = 6 cos x, which leads to cos x = frac{2}{3}. Using the fact that P_1 lies on the x-axis, P_1 will have the same x-coordinate as P but with a y-coordinate of 0. Therefore, the x-coordinate of point P (and P_1) is obtained from the relation cos x = frac{2}{3} within the given interval (0, frac{pi}{2}). Now, to find the length of the line segment P_1P_2, we need to find the y-coordinate of point P_2 by substituting the x-coordinate of P_1 into the equation y = cos x: y = cos x = boxed{frac{2}{3}}. Thus, the length of P_1P_2 is boxed{frac{2}{3}} because the y-coordinate of P_1 is 0 and the y-coordinate of P_2 is the value of the cosine function at the determined x-coordinate.