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question:Let m and n be two different lines, and alpha, beta, gamma be three different planes. Given the following four propositions: ① If m perp alpha and n parallel alpha, then m perp n; ② If alpha perp gamma, beta perp gamma, and alpha cap beta = m, then m perp gamma; ③ If m parallel alpha and n subset alpha, then m parallel n; ④ If alpha perp beta, alpha cap beta = n, and m perp n, then m perp beta Among these propositions, the correct ones are ______.
answer:For proposition ①, since n parallel alpha, we can construct a plane beta through n such that beta cap alpha = l, which leads to n parallel l. Since m perp alpha and l subset alpha, it follows that m perp l. Combining this with n parallel l yields m perp n. Therefore, proposition ① is true. For proposition ②, since alpha and beta are both perpendicular to the same plane gamma, the intersection line of alpha and beta must be perpendicular to gamma. This makes proposition ② true. For proposition ③, if m parallel alpha and n subset alpha, then m could be parallel to n or skew, which makes proposition ③ false. For proposition ④, if alpha perp beta, alpha cap beta = n, and m perp n with m subset alpha, then m perp beta, which is a false proposition. Therefore, the answer is: boxed{text{①②}}. Based on the theorem of line-plane parallelism and the definition of line-plane perpendicularity, proposition ① is determined to be true; By judging that if two planes are both perpendicular to the same plane, then their intersection line must be perpendicular to the third plane, proposition ② is determined to be true; Counterexamples can be provided for propositions ③ and ④. This question presents propositions about the spatial relationships between lines and planes, asking us to identify the true propositions. It focuses on the properties of line-plane parallelism, plane-plane parallelism, line-plane perpendicularity, and plane-plane perpendicularity, making it a medium-difficulty question.
question:During a mega sale event, Daniela went on a shopping spree. The store was offering 25% off on shoes, 30% off on dresses, 40% off on bags, 15% off on accessories and an additional 10% off on the total bill if the total cost before applying this discount exceeds 350. On top of that, a sales tax of 8% was applicable on the discounted price. If Daniela bought 4 pairs of shoes originally priced at 70 each, 2 dresses originally priced at 150 each, a handbag originally priced at 80, and 5 accessories originally priced at 20 each, how much does she end up spending after all the discounts and including the sales tax?
answer:First, let's calculate the total cost of each item category before any discounts: Shoes: 4 pairs * 70 = 280 Dresses: 2 dresses * 150 = 300 Handbag: 1 bag * 80 = 80 Accessories: 5 accessories * 20 = 100 Now, let's apply the respective discounts to each category: Discount on shoes: 25% of 280 = 0.25 * 280 = 70 Discount on dresses: 30% of 300 = 0.30 * 300 = 90 Discount on handbag: 40% of 80 = 0.40 * 80 = 32 Discount on accessories: 15% of 100 = 0.15 * 100 = 15 Subtract the discounts from the original prices to get the discounted prices: Discounted price of shoes: 280 - 70 = 210 Discounted price of dresses: 300 - 90 = 210 Discounted price of handbag: 80 - 32 = 48 Discounted price of accessories: 100 - 15 = 85 Now, let's calculate the total cost after these discounts: Total cost after discounts = 210 (shoes) + 210 (dresses) + 48 (handbag) + 85 (accessories) = 553 Since the total cost before the additional 10% discount exceeds 350, Daniela gets an additional 10% off: Additional 10% discount = 10% of 553 = 0.10 * 553 = 55.30 Subtract this additional discount from the total cost after discounts: Total cost after all discounts = 553 - 55.30 = 497.70 Finally, we need to add the sales tax of 8% to this amount: Sales tax = 8% of 497.70 = 0.08 * 497.70 = 39.816 Adding the sales tax to the total cost after all discounts: Total cost including sales tax = 497.70 + 39.816 = 537.516 Rounding to the nearest cent, Daniela ends up spending boxed{537.52} after all the discounts and including the sales tax.
question:The average (arithmetic mean) of the integers from 200 to 400, inclusive, is how much greater than the average of the integers from 100 to 200, inclusive?
answer:To find the average of a set of consecutive integers, you can take the average of the first and last numbers in the set. For the integers from 200 to 400, inclusive: First number (F1) = 200 Last number (L1) = 400 Average (A1) = (F1 + L1) / 2 = (200 + 400) / 2 = 600 / 2 = 300 For the integers from 100 to 200, inclusive: First number (F2) = 100 Last number (L2) = 200 Average (A2) = (F2 + L2) / 2 = (100 + 200) / 2 = 300 / 2 = 150 Now, to find how much greater the average of the integers from 200 to 400 is than the average of the integers from 100 to 200: Difference = A1 - A2 = 300 - 150 = 150 The average of the integers from 200 to 400 is boxed{150} greater than the average of the integers from 100 to 200.
question:For (x > 0), the area of the triangle with vertices ((0, 0), (x, 3x)), and ((x, 0)) is 96 square units. What is the value of (x)?
answer:1. **Plot the Points and Identify the Triangle**: The vertices given are ((0, 0)), ((x, 3x)), and ((x, 0)). This forms a right triangle with the right angle at ((x, 0)). 2. **Apply the Area Formula**: The base of the triangle is along the x-axis from ((0, 0)) to ((x, 0)) and measures (x) units. The height of the triangle from ((x, 0)) to ((x, 3x)) is (3x) units. The area of the triangle is given by [ text{Area} = frac{1}{2} times text{base} times text{height} = frac{1}{2} times x times 3x = frac{3x^2}{2}. ] Setting this equal to 96 square units, [ frac{3x^2}{2} = 96 implies 3x^2 = 192 implies x^2 = 64 implies x = 8. ] Since (x > 0), we take the positive root. 3. **Conclusion**: The value of (x) that satisfies the condition is (boxed{8}) units.