answer:To find p(a ∩ b ∩ c), which is the probability of all three events a, b, and c occurring together, we can simply multiply the individual probabilities together because the events are independent. p(a ∩ b ∩ c) = p(a) * p(b) * p(c) p(a ∩ b ∩ c) = (5/7) * (2/5) * (3/4) Now, let's calculate the product: p(a ∩ b ∩ c) = (5 * 2 * 3) / (7 * 5 * 4) p(a ∩ b ∩ c) = 30 / 140 p(a ∩ b ∩ c) = 3 / 14 So, the probability of all three events a, b, and c occurring together is 3/14. Next, to find p(a ∩ b | c), which is the probability of events a and b occurring given that event c has occurred, we can use the definition of conditional probability. Since events a and b are independent, their intersection given c is simply the product of their individual probabilities: p(a ∩ b | c) = p(a | c) * p(b | c) Because a and b are independent of c, p(a | c) = p(a) and p(b | c) = p(b). Therefore: p(a ∩ b | c) = p(a) * p(b) p(a ∩ b | c) = (5/7) * (2/5) Now, let's calculate the product: p(a ∩ b | c) = (5 * 2) / (7 * 5) p(a ∩ b | c) = 10 / 35 p(a ∩ b | c) = 2 / 7 So, the probability of events a and b occurring given that event c has occurred is boxed{2/7} .

answer:Let A(x_1, y_1) and B(x_2, y_2), |AB|=x_{1}+x_{2}+p= frac {2p}{sin^{2}theta }= frac {8p}{3}, x_{1}+x_{2}= frac {5p}{3}, Since x_{1}x_{2}= frac {p^{2}}{4}, we can find x_{1}= frac {3}{2}p, x_{2}= frac {p}{6}, Thus, frac {|AF|}{|BF|}= frac { frac {3p}{2}+ frac {p}{2}}{ frac {p}{2}+ frac {p}{6}}=3, Therefore, the correct answer is boxed{C}. By setting the coordinates for A and B, and using the formula for the focal radius to find |AB|, combined with x_{1}x_{2}= frac {p^{2}}{4}, we can find the coordinates for A and B, and then calculate their ratio. This question tests the inclination angle of a line, simple properties of a parabola, and the student's ability to analyze and solve problems. It is a basic question.

answer:Let's denote the total amount of all the items before sales tax as ( x ). The sales tax is 8% of the total amount purchased, so the sales tax amount would be ( 0.08x ). The total amount Pauline will spend, including sales tax, is the sum of the amount before sales tax and the sales tax amount, which is ( x + 0.08x ). We know that this total amount including sales tax is 162, so we can set up the following equation: ( x + 0.08x = 162 ) Combining like terms, we get: ( 1.08x = 162 ) Now, we can solve for ( x ) by dividing both sides of the equation by 1.08: ( x = frac{162}{1.08} ) ( x = 150 ) So, the total amount of all the items before sales tax is boxed{150} .

answer:1. **Calculate the perimeter of the rectangular patio**: The formula for the perimeter (P) of a rectangle is given by (P = 2(l + w)), where (l) is the length and (w) is the width of the rectangle. For a patio with length (24) feet and width (10) feet, the perimeter is: [ P = 2(24 + 10) = 2 times 34 = 68 text{ feet} ] 2. **Determine the number of bushes**: Since the bushes are spaced approximately 1.2 feet apart, the number of bushes needed is approximately equal to the perimeter divided by the spacing between bushes. Thus: [ text{Number of bushes} = frac{68}{1.2} approx 56.67 ] Rounding 56.67 to the nearest whole number, we get: [ text{Number of bushes} approx 57 ] 3. **Conclusion**: The rounded number of bushes required to surround the rectangular patio is 57. The final answer is boxed{text{C}}