💡Alright, so I've got this problem here about a scuba diver descending to a ship that's lost at a depth of 3600 feet below sea level. The diver takes 60 minutes to reach the ship, and I need to find out the rate at which the diver descends in feet per minute. Hmm, okay, let's break this down step by step.First off, I know that rate is generally calculated as distance divided by time. In this case, the distance is the depth the diver descends, which is 3600 feet, and the time is 60 minutes. So, the basic formula I should use is:Rate = Distance / TimePlugging in the numbers I have:Rate = 3600 feet / 60 minutesNow, doing that division, 3600 divided by 60. Let me see, 60 times 60 is 3600, so 3600 divided by 60 should be 60. So, does that mean the diver is descending at 60 feet per minute?Wait a second, that seems pretty fast. I thought scuba divers usually don't descend that quickly. Maybe I should double-check my calculations. Let me try dividing 3600 by 60 again.3600 divided by 60 equals 60. Yeah, that's correct mathematically. But is 60 feet per minute a realistic descent rate for a scuba diver? I think the standard recommendation is to descend no faster than 60 feet per minute to avoid issues like barotrauma or decompression sickness. So, in this case, the diver is descending at the maximum recommended rate.Okay, so my calculation seems to align with the standard guidelines. Therefore, the rate at which the diver descends is indeed 60 feet per minute.Just to make sure I'm not missing anything, let me consider if there are any other factors that might affect the descent rate, like water currents or the diver's buoyancy. But the problem doesn't mention any of these, so I think it's safe to assume that the descent is at a constant rate without any external influences.Also, I should confirm the units to ensure consistency. The depth is given in feet, and the time is in minutes, so the rate will naturally be in feet per minute, which is what the problem is asking for.In summary, by dividing the total depth by the total time, I've calculated that the diver descends at a rate of 60 feet per minute, which is both mathematically correct and aligns with standard diving practices.

💡First, I need to understand that inserting two numbers between -1 and 8 will create an arithmetic sequence of four numbers.In an arithmetic sequence, the difference between consecutive terms is constant. This constant difference is called the common difference, denoted as ( d ).Since there are two numbers inserted between -1 and 8, the sequence will have four terms. The first term is -1, and the fourth term is 8.To find the common difference ( d ), I can use the formula for the ( n )-th term of an arithmetic sequence:[a_n = a_1 + (n-1)d]Here, ( a_1 = -1 ), ( a_4 = 8 ), and ( n = 4 ). Plugging in these values:[8 = -1 + (4-1)d][8 = -1 + 3d][3d = 9][d = 3]Now that I have the common difference ( d = 3 ), I can find the two numbers ( a ) and ( b ) to be inserted between -1 and 8.The second term ( a ) is:[a = a_1 + d = -1 + 3 = 2]The third term ( b ) is:[b = a + d = 2 + 3 = 5]Therefore, the two numbers to be inserted are ( a = 2 ) and ( b = 5 ).

💡First, I'll denote the charges for a single room at hotels G, R, and P as G, R, and P respectively.From the problem, I know that:1. P is 10% less than G, so P = 0.90G.2. R is 12.500000000000004% greater than G, so R = 1.125G.To find the percentage difference between P and R, I'll express P in terms of R.Since R = 1.125G, I can solve for G:G = R / 1.125Substituting this back into the equation for P:P = 0.90 * (R / 1.125) = 0.80RThis means P is 80% of R. To find the percentage difference:Percentage Difference = 100% - 80% = 20%Therefore, the percentage difference between the charges for a single room at hotel P and hotel R is 20%.

💡Okay, so I need to figure out how much Gordon will spend on books after applying the discounts, tax, and extra fees. Let's see, there are 12 books he's buying, each with different prices and corresponding discounts. I need to calculate the total cost step by step.First, I'll list out all the books with their prices and the discounts they qualify for:1. 25.00 - Discount A (30% off)2. 18.00 - Discount B (20% off)3. 21.00 - Discount B (20% off)4. 35.00 - Discount A (30% off)5. 12.00 - Discount B (20% off)6. 10.00 - Discount C (15% off)7. 8.50 - Discount C (15% off)8. 23.00 - Discount A (30% off)9. 6.00 - Discount C (15% off)10. 15.50 - Discount B (20% off)11. 30.00 - Discount A (30% off)12. 9.50 - Discount C (15% off)Alright, so I need to apply each discount to the respective book prices. Let's start with Discount A, which is 30% off for books over 22.00.For Discount A:1. 25.00 - 30% = 25.00 * 0.70 = 17.504. 35.00 - 30% = 35.00 * 0.70 = 24.508. 23.00 - 30% = 23.00 * 0.70 = 16.1011. 30.00 - 30% = 30.00 * 0.70 = 21.00Adding these up: 17.50 + 24.50 + 16.10 + 21.00 = 79.10Next, Discount B, which is 20% off for books between 10.01 and 20.00.For Discount B:2. 18.00 - 20% = 18.00 * 0.80 = 14.403. 21.00 - 20% = 21.00 * 0.80 = 16.805. 12.00 - 20% = 12.00 * 0.80 = 9.6010. 15.50 - 20% = 15.50 * 0.80 = 12.40Adding these up: 14.40 + 16.80 + 9.60 + 12.40 = 53.20Now, Discount C, which is 15% off for books under 10.00.For Discount C:6. 10.00 - 15% = 10.00 * 0.85 = 8.507. 8.50 - 15% = 8.50 * 0.85 = 7.239. 6.00 - 15% = 6.00 * 0.85 = 5.1012. 9.50 - 15% = 9.50 * 0.85 = 8.08Adding these up: 8.50 + 7.23 + 5.10 + 8.08 = 28.91Now, let's sum up the totals from each discount category:Total after discounts = 79.10 (A) + 53.20 (B) + 28.91 (C) = 161.21Next, we need to apply the tax rate of 5% to this total.Tax = 161.21 * 0.05 = 8.06Adding the tax to the total after discounts:Total with tax = 161.21 + 8.06 = 169.27Finally, we add the extra fee of 2.00:Total with tax and extra fee = 169.27 + 2.00 = 171.27So, Gordon will spend 171.27 on books after applying the discounts, tax, and extra fees.

💡Okay, so I've got this problem here: If ( a(x+2) + b(x+2) = 60 ) and ( a + b = 12 ), then ( x ) is equal to: (A) 3 (B) 5 (C) 1 (D) 7 (E) 48Alright, let's see. I need to find the value of ( x ). Hmm, okay, let's break this down step by step.First, looking at the equation ( a(x+2) + b(x+2) = 60 ). I notice that both terms on the left side have ( (x+2) ) as a common factor. Maybe I can factor that out to simplify the equation. Let me try that.So, factoring out ( (x+2) ), the equation becomes:[ (a + b)(x + 2) = 60 ]Oh, that's nice because I also have another equation ( a + b = 12 ). Maybe I can substitute that into the equation I just got. Let me do that.Substituting ( a + b = 12 ) into the equation:[ 12(x + 2) = 60 ]Okay, now I have a simpler equation. I need to solve for ( x ). Let's divide both sides of the equation by 12 to isolate ( (x + 2) ).Dividing both sides by 12:[ x + 2 = frac{60}{12} ][ x + 2 = 5 ]Now, to solve for ( x ), I just need to subtract 2 from both sides.Subtracting 2 from both sides:[ x = 5 - 2 ][ x = 3 ]So, ( x ) equals 3. Let me check the options given. Option (A) is 3, so that must be the correct answer.Wait, just to make sure I didn't make any mistakes. Let me go through the steps again.1. Original equation: ( a(x+2) + b(x+2) = 60 ).2. Factor out ( (x+2) ): ( (a + b)(x + 2) = 60 ).3. Substitute ( a + b = 12 ): ( 12(x + 2) = 60 ).4. Divide both sides by 12: ( x + 2 = 5 ).5. Subtract 2: ( x = 3 ).Yep, that all checks out. I don't see any errors in my reasoning. So, I'm confident that ( x = 3 ) is the correct answer.I think I'm done here. It wasn't too bad once I factored out the common term. Sometimes, recognizing common factors can simplify the problem a lot. I should remember that for future problems.

💡First, I need to determine the denomination of the bills produced by Press T. Let's denote this denomination as ( x ) dollars.Press F produces 5-dollar bills at a rate of 1,000 bills per minute. To find the production rate per second, I divide by 60:[frac{1000}{60} text{ bills per second}]The value of currency produced by Press F per second is:[5 times frac{1000}{60} = frac{5000}{60} text{ dollars per second}]Similarly, Press T produces ( x )-dollar bills at a rate of 200 bills per minute. Converting this to bills per second:[frac{200}{60} text{ bills per second}]The value of currency produced by Press T per second is:[x times frac{200}{60} = frac{200x}{60} text{ dollars per second}]According to the problem, it takes 3 seconds for Press F to produce 50 more currency than Press T. Setting up the equation:[3 times left( frac{5000}{60} right) = 3 times left( frac{200x}{60} right) + 50]Simplifying both sides:[frac{15000}{60} = frac{600x}{60} + 50][250 = 10x + 50]Subtracting 50 from both sides:[200 = 10x]Dividing both sides by 10:[x = 20]Therefore, the denomination of the bills produced by Press T is 20.