answer：Okay, so I have this problem here: Given that the angle between the non-coplanar 3D vectors **u** and **v** is 30 degrees, what is the angle between the vectors 2**u** and -**v**? Hmm, let me try to figure this out step by step.First, I remember that the angle between two vectors can be found using the dot product formula. The formula is:[cos{theta} = frac{mathbf{u} cdot mathbf{v}}{||mathbf{u}|| , ||mathbf{v}||}]where (theta) is the angle between vectors **u** and **v**. In this case, we're given that (theta = 30^circ).Now, the problem is asking about the angle between 2**u** and -**v**. Let me think about how scaling and negating vectors affects the angle between them.Scaling a vector by a positive scalar (like 2 in this case) doesn't change the direction of the vector, only its magnitude. So, 2**u** is just a longer version of **u** in the same direction. On the other hand, negating a vector (like -**v**) flips its direction. So, -**v** points exactly opposite to **v**.I think this means that the angle between 2**u** and -**v** is related to the original angle between **u** and **v**. Since -**v** is in the opposite direction of **v**, the angle between **u** and -**v** should be 180 degrees minus the original angle between **u** and **v**.Let me verify this. If the original angle is 30 degrees, then the angle between **u** and -**v** should be 180 - 30 = 150 degrees. But wait, we're dealing with 2**u** instead of **u**. Does scaling affect the angle?No, scaling a vector by a positive scalar doesn't change the angle between vectors because it only affects the magnitude, not the direction. So, even though 2**u** is twice as long as **u**, it still points in the same direction. Therefore, the angle between 2**u** and -**v** should still be 150 degrees.Just to make sure, let me use the dot product formula again with the scaled and negated vectors. The dot product of 2**u** and -**v** is:[(2mathbf{u}) cdot (-mathbf{v}) = -2 (mathbf{u} cdot mathbf{v})]The magnitudes of 2**u** and -**v** are:[||2mathbf{u}|| = 2||mathbf{u}||][||-mathbf{v}|| = ||mathbf{v}||]So, plugging into the cosine formula:[cos{phi} = frac{-2 (mathbf{u} cdot mathbf{v})}{2||mathbf{u}|| , ||mathbf{v}||} = frac{- (mathbf{u} cdot mathbf{v})}{||mathbf{u}|| , ||mathbf{v}||} = -cos{30^circ}]Since (cos{phi} = -cos{30^circ}), this implies that (phi = 180^circ - 30^circ = 150^circ). Yep, that checks out.So, the angle between 2**u** and -**v** is 150 degrees.

answer：Okay, so I have this problem about a league with 50 teams, and each team plays against every other team four times during the season. I need to figure out how many total games are played in the season. Hmm, let's see. First, I think I need to understand how many unique pairings there are between the teams. Since there are 50 teams, each team will play against 49 other teams. But if I just multiply 50 by 49, that would give me the total number of games from each team's perspective, but that would count each game twice because, for example, Team A vs Team B is the same game as Team B vs Team A. So, to get the actual number of unique pairings, I should divide that number by 2.So, the number of unique pairings would be (50 * 49) / 2. Let me calculate that: 50 times 49 is 2450, and dividing that by 2 gives me 1225 unique pairings.Now, since each pair of teams plays each other four times, I need to multiply the number of unique pairings by 4 to get the total number of games. So, 1225 times 4 equals 4900 games.Wait, does that make sense? Let me double-check. If there are 50 teams, each playing 49 others four times, that's 50 * 49 * 4. But that would count each game twice, right? So, to get the actual number of games, I should divide by 2. So, (50 * 49 * 4) / 2. Let me calculate that: 50 times 49 is 2450, times 4 is 9800, divided by 2 is 4900. Yeah, that matches what I got before.So, it seems like the total number of games played in the season is 4900.

answer：Okay, so I need to figure out how many integers from 1 to 9 are divisors of the number 46,320. Hmm, that sounds like a problem where I have to check each number from 1 to 9 and see if 46,320 is divisible by them. I remember there are some rules for checking divisibility without actually doing the full division, which might save me some time.Let me start with the basics. First, every number is divisible by 1, right? So, 1 is definitely a divisor. That's easy.Next, checking for divisibility by 2. I recall that if a number is even, meaning its last digit is even (0, 2, 4, 6, or 8), then it's divisible by 2. Looking at 46,320, the last digit is 0, which is even. So, 2 is also a divisor.Moving on to 3. The rule for 3 is that if the sum of all the digits of the number is divisible by 3, then the number itself is divisible by 3. Let's add up the digits of 46,320: 4 + 6 + 3 + 2 + 0. That gives me 15. Now, 15 divided by 3 is 5, which is a whole number, so 3 is a divisor.For 4, the rule is a bit different. A number is divisible by 4 if the number formed by its last two digits is divisible by 4. The last two digits of 46,320 are 20. 20 divided by 4 is 5, so that works. Therefore, 4 is a divisor.Next up is 5. The rule for 5 is straightforward: if a number ends in 0 or 5, it's divisible by 5. Since 46,320 ends with a 0, it's definitely divisible by 5.Now, checking for 6. I remember that a number is divisible by 6 if it's divisible by both 2 and 3. We've already established that 46,320 is divisible by both 2 and 3, so it must also be divisible by 6.Moving on to 7. This one is a bit trickier. I think the rule for 7 involves doubling the last digit and subtracting it from the rest of the number, then seeing if the result is divisible by 7. Let's try that. The last digit is 0, so doubling it gives 0. Subtracting that from the remaining number, which is 4632, gives 4632 - 0 = 4632. Hmm, that didn't change anything. Maybe I need to repeat the process. Taking 4632, the last digit is 2. Doubling it gives 4. Subtracting that from 463 gives 463 - 4 = 459. Now, take 459. The last digit is 9. Doubling it gives 18. Subtracting that from 45 gives 45 - 18 = 27. 27 isn't divisible by 7, so 46,320 isn't divisible by 7.Next is 8. The rule for 8 is similar to that of 4 but with the last three digits. If the number formed by the last three digits is divisible by 8, then the whole number is divisible by 8. The last three digits of 46,320 are 320. Dividing 320 by 8 gives 40, which is a whole number. So, 8 is a divisor.Finally, checking for 9. The rule for 9 is similar to that of 3, but the sum of the digits needs to be divisible by 9. Earlier, we found that the sum of the digits is 15. 15 divided by 9 is 1.666..., which isn't a whole number. Therefore, 46,320 isn't divisible by 9.So, to recap, the numbers from 1 to 9 that divide 46,320 are 1, 2, 3, 4, 5, 6, and 8. That makes a total of 7 numbers.

answer：First, I observe that the expression involves multiplying two powers of the same base, (frac{9}{11}), with exponents 4 and -4.I recall the law of exponents which states that when multiplying powers with the same base, we add their exponents: (a^n cdot a^m = a^{n+m}).Applying this law to the given expression:[left(frac{9}{11}right)^4 cdot left(frac{9}{11}right)^{-4} = left(frac{9}{11}right)^{4 + (-4)} = left(frac{9}{11}right)^0]I know that any non-zero number raised to the power of 0 is equal to 1:[left(frac{9}{11}right)^0 = 1]Therefore, the final answer is 1.

answer：Okay, so I need to find the molecular weight of 3 moles of HBrO3. Hmm, where do I start? I remember that molecular weight is the sum of the atomic weights of all the atoms in a molecule. So, first, I should figure out the molecular weight of one mole of HBrO3, and then multiply that by 3 to get the weight for 3 moles.Alright, let's break down the chemical formula HBrO3. It has one hydrogen atom (H), one bromine atom (Br), and three oxygen atoms (O). I think I need to look up the atomic weights of these elements. I recall that hydrogen has an atomic weight of about 1.01 g/mol, bromine is around 79.90 g/mol, and oxygen is approximately 16.00 g/mol.So, for one mole of HBrO3, the molecular weight would be:1.01 (for H) + 79.90 (for Br) + 3 * 16.00 (for O).Let me calculate that:1.01 + 79.90 = 80.91.Then, 3 * 16.00 = 48.00.Adding those together: 80.91 + 48.00 = 128.91 g/mol.Okay, so one mole of HBrO3 weighs 128.91 grams. Now, to find the weight of 3 moles, I just multiply this by 3.128.91 * 3 = ?Let me do that multiplication step by step:128.91 * 3:128 * 3 = 384.0.91 * 3 = 2.73.Adding those together: 384 + 2.73 = 386.73 grams.So, 3 moles of HBrO3 would weigh 386.73 grams.Wait, does that make sense? Let me double-check my calculations.Hydrogen: 1.01 g/mol.Bromine: 79.90 g/mol.Oxygen: 16.00 g/mol * 3 = 48.00 g/mol.Adding them up: 1.01 + 79.90 + 48.00 = 128.91 g/mol.Yes, that's correct for one mole. Then, 128.91 * 3 = 386.73 grams.I think that's right. But just to be sure, maybe I can check the molecular weight of HBrO3 somewhere else or see if there's a standard value for it. But based on my calculations, it seems accurate.

answer：First, I need to determine the volume of the first cylindrical container. The formula for the volume of a cylinder is ( V = pi r^2 h ), where ( r ) is the radius and ( h ) is the height. The first container has a diameter of 2 inches, so the radius is 1 inch, and a height of 5 inches. Plugging these values into the formula, the volume is ( V_1 = pi times 1^2 times 5 = 5pi ) cubic inches.Next, I'll calculate the volume of the second container. It has a diameter of 4 inches, so the radius is 2 inches, and a height of 10 inches. Using the same volume formula, the volume is ( V_2 = pi times 2^2 times 10 = 40pi ) cubic inches.Now, I need to find the price of the second container based on the given rate. The first container costs 0.75 and has a volume of ( 5pi ) cubic inches. The second container has a volume of ( 40pi ) cubic inches, which is 8 times the volume of the first container (( frac{40pi}{5pi} = 8 )). Therefore, the price of the second container should be 8 times the price of the first container, which is ( 0.75 times 8 = 6.00 ).