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question:Given the ellipse C: frac{x^{2}}{a^{2}}+ frac{y^{2}}{2}=1 (a > sqrt{2}), its left foci are F_1 and F_2, and its eccentricity is e. Line l: y=ex+, and point P is the point symmetrical to F_1 about line l. If triangle PF_2 is an isosceles triangle, then find the value of a.
answer:According to the problem, we can derive that c = sqrt{a^{2}-2} and e = frac{sqrt{a^{2}-2}}{a}. Solving for a, we get a = sqrt{3}. The distance d between the point (-c, 0) and the line l is given by d = frac{|-ec|}{sqrt{1+e^{2}}}. Simplifying, we get 4 - 3sqrt{2} = 0, which yields 2d = c. Substituting the values of d and c and solving for a, we get a = sqrt{3}. Utilizing the formula for the eccentricity of an ellipse and the relationship between a, b, and c, along with the formula for the distance between a point and a line, we can solve for a. Thus, the answer is boxed{a = sqrt{3}}.
question:Jeanne wants to ride the Ferris wheel, the roller coaster, and the bumper cars. The Ferris wheel costs 5 tickets, the roller coaster costs 4 tickets and the bumper cars cost 4 tickets. Jeanne has 5 tickets. How many more tickets should Jeanne buy?
answer:Jeanne wants to ride three attractions: the Ferris wheel (5 tickets), the roller coaster (4 tickets), and the bumper cars (4 tickets). The total number of tickets she needs for all three rides is: 5 (Ferris wheel) + 4 (roller coaster) + 4 (bumper cars) = 13 tickets Jeanne already has 5 tickets, so she needs to buy additional tickets to have enough for all three rides. The number of additional tickets she needs is: 13 (total tickets needed) - 5 (tickets she already has) = 8 tickets Therefore, Jeanne should buy boxed{8} more tickets.
question:Express frac{7}{16} as a decimal.
answer:To convert frac{7}{16} to a decimal, we follow similar steps as the original problem: 1. frac{7}{16} can be expressed with a denominator that is a power of 10 by recognizing that the denominator, 16, is 2^4. We multiply both the numerator and the denominator by 5^4 = 625: [ frac{7}{16} = frac{7 times 625}{16 times 625} = frac{4375}{10000}. ] 2. This fraction simplifies directly to a decimal: [ frac{4375}{10000} = 0.4375. ] 3. Thus, frac{7}{16} as a decimal is boxed{0.4375}.
question:a set of four consecutive integers, the largest integer is less than three times the smallest integer minus five. What is the smallest integer in the set?
answer:Let the integers be m, m+1, m+2, and m+3. The smallest integer is m and the largest is m+3. According to the problem's statement, the inequality to solve is: m+3 < 3m - 5 Solving for m: [ begin{align*} m + 3 &< 3m - 5 3 + 5 &< 3m - m 8 &< 2m 4 &< m end{align*} ] Thus, m must be greater than 4. Since m must be an integer, the smallest possible value for m is boxed{5}.