💡First, I need to calculate the income generated from each of the hotel's services: meetings, events, and rooms. The payments for these services are given as fractions of the total operations expenses.1. **Meetings Income**: [ frac{5}{8} times 5,000 = 3,125 ]2. **Events Income**: [ frac{3}{10} times 5,000 = 1,500 ]3. **Rooms Income**: [ frac{11}{20} times 5,000 = 2,750 ]Next, I'll calculate the taxes and employee commissions for each service based on the respective percentages provided.1. **Meetings**: - **Tax**: (10%) of 3,125 [ 0.10 times 3,125 = 312.50 ] - **Commission**: (5%) of 3,125 [ 0.05 times 3,125 = 156.25 ] - **Total Additional Expenses for Meetings**: [ 312.50 + 156.25 = 468.75 ]2. **Events**: - **Tax**: (8%) of 1,500 [ 0.08 times 1,500 = 120 ] - **Commission**: (6%) of 1,500 [ 0.06 times 1,500 = 90 ] - **Total Additional Expenses for Events**: [ 120 + 90 = 210 ]3. **Rooms**: - **Tax**: (12%) of 2,750 [ 0.12 times 2,750 = 330 ] - **Commission**: (3%) of 2,750 [ 0.03 times 2,750 = 82.50 ] - **Total Additional Expenses for Rooms**: [ 330 + 82.50 = 412.50 ]Now, I'll sum up all the additional expenses:[468.75 , (text{Meetings}) + 210 , (text{Events}) + 412.50 , (text{Rooms}) = 1,091.25]Next, I'll calculate the total income from all services:[3,125 , (text{Meetings}) + 1,500 , (text{Events}) + 2,750 , (text{Rooms}) = 7,375]Finally, to determine the total profit or loss, I'll subtract the total operations expenses and the additional expenses from the total income:[text{Total Profit or Loss} = 7,375 - (5,000 + 1,091.25) = 7,375 - 6,091.25 = 1,283.75]

💡First, I recognize that (cos 78^circ) is given as approximately 0.20. I want to find (sin 66^circ).I recall the co-function identity that relates cosine and sine: (cos theta = sin(90^circ - theta)). Applying this identity, I can express (cos 78^circ) as (sin(12^circ)), since (90^circ - 78^circ = 12^circ). Therefore, (sin 12^circ = 0.20).Next, I need to find (sin 66^circ). I can use another co-function identity: (sin theta = cos(90^circ - theta)). Applying this, (sin 66^circ = cos(24^circ)), since (90^circ - 66^circ = 24^circ).Now, I need to find (cos 24^circ). I can use the double-angle formula for cosine, which states that (cos 2theta = 1 - 2sin^2 theta). Letting (theta = 12^circ), I have:[cos 24^circ = 1 - 2sin^2 12^circ]Substituting the given value of (sin 12^circ = 0.20):[cos 24^circ = 1 - 2(0.20)^2 = 1 - 2(0.04) = 1 - 0.08 = 0.92]Therefore, (sin 66^circ = cos 24^circ = 0.92).

💡Okay, so I have this problem here: "How many integer values of ( x ) satisfy ( |x| < 5pi )?" And the options are A) 29, B) 30, C) 31, D) 32, E) 33. Hmm, let me try to figure this out step by step.First, I know that the absolute value inequality ( |x| < a ) means that ( x ) is between ( -a ) and ( a ). So in this case, ( |x| < 5pi ) translates to ( -5pi < x < 5pi ). That makes sense.Now, I need to find how many integer values of ( x ) satisfy this inequality. To do that, I should figure out the range of integers between ( -5pi ) and ( 5pi ). But wait, ( pi ) is approximately 3.1416, right? So ( 5pi ) would be about ( 5 times 3.1416 ), which is roughly 15.708.So, ( 5pi ) is approximately 15.708. That means the inequality ( -15.708 < x < 15.708 ). Since ( x ) has to be an integer, the smallest integer greater than ( -15.708 ) is ( -15 ) and the largest integer less than ( 15.708 ) is ( 15 ).Therefore, the integers that satisfy this inequality are from ( -15 ) to ( 15 ), inclusive. Now, I need to count how many integers are in this range.Let me think: from ( -15 ) to ( 15 ), how many numbers is that? Well, if I consider the negative numbers, zero, and the positive numbers, I can count them separately.Starting from ( -15 ) up to ( -1 ), that's 15 numbers. Then, including zero, that's 16 numbers. Then from ( 1 ) to ( 15 ), that's another 15 numbers. So total numbers would be ( 15 + 1 + 15 = 31 ).Wait, let me double-check that. If I list them out: ( -15, -14, -13, ldots, -1, 0, 1, 2, ldots, 14, 15 ). So from ( -15 ) to ( 15 ), how many terms are there?Another way to calculate the number of integers between two numbers is to take the difference and add 1. So, from ( -15 ) to ( 15 ), the difference is ( 15 - (-15) = 30 ). Then, adding 1 gives ( 31 ) integers. Yeah, that seems right.But wait, let me make sure I didn't make a mistake in my initial approximation. ( 5pi ) is approximately 15.708, so the integers less than that are up to 15, and the integers greater than ( -15.708 ) are down to ( -15 ). So, yes, the integers from ( -15 ) to ( 15 ) inclusive are all within the range ( -15.708 < x < 15.708 ).So, counting them, as I did before, gives 31 integers. Therefore, the correct answer should be C) 31.But just to be thorough, let me consider if there's any chance I might have miscounted. Sometimes when counting inclusive ranges, it's easy to make an off-by-one error. So, let's see: from ( -15 ) to ( 15 ), how many numbers is that?If I consider the positive numbers: 1 to 15 is 15 numbers. The negative numbers: -1 to -15 is another 15 numbers. Then, including zero, that's 1 more. So, 15 + 15 + 1 = 31. Yep, that's correct.Alternatively, using the formula for the number of integers from ( a ) to ( b ) inclusive, which is ( b - a + 1 ). Here, ( a = -15 ) and ( b = 15 ). So, ( 15 - (-15) + 1 = 15 + 15 + 1 = 31 ). That confirms it.I think I'm confident now that the answer is 31, which is option C.

💡Okay, so I need to find the value of ( pq + pr + qr ) where ( p ), ( q ), and ( r ) are the roots of the cubic equation ( 6x^3 - 5x^2 + 20x - 10 = 0 ). Hmm, I remember that there are some formulas related to the roots of polynomials, called Vieta's formulas. Let me try to recall how they work.Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. For a cubic equation of the form ( ax^3 + bx^2 + cx + d = 0 ), the sum of the roots ( p + q + r ) is equal to ( -frac{b}{a} ). The sum of the products of the roots taken two at a time, which is ( pq + pr + qr ), is equal to ( frac{c}{a} ). And the product of all three roots ( pqr ) is equal to ( -frac{d}{a} ).So, in my case, the given polynomial is ( 6x^3 - 5x^2 + 20x - 10 = 0 ). Comparing this with the general form ( ax^3 + bx^2 + cx + d = 0 ), I can identify the coefficients:- ( a = 6 )- ( b = -5 )- ( c = 20 )- ( d = -10 )I need to find ( pq + pr + qr ), which according to Vieta's formula is ( frac{c}{a} ). Plugging in the values, that would be ( frac{20}{6} ). Wait, that simplifies to ( frac{10}{3} ). So, is that the answer?Let me double-check to make sure I didn't make a mistake. Vieta's formula says that ( pq + pr + qr = frac{c}{a} ). Here, ( c = 20 ) and ( a = 6 ), so yes, ( frac{20}{6} ) simplifies to ( frac{10}{3} ). That seems right.Just to be thorough, let me recall why Vieta's formulas work. For a cubic equation, if you expand ( (x - p)(x - q)(x - r) ), you get ( x^3 - (p + q + r)x^2 + (pq + pr + qr)x - pqr ). When you compare this with the given polynomial ( 6x^3 - 5x^2 + 20x - 10 ), you can see that the coefficients correspond to the sums and products of the roots, scaled by the leading coefficient.So, in the expanded form, the coefficient of ( x^2 ) is ( -(p + q + r) ), which matches ( -5 ) when scaled by ( 6 ). Similarly, the coefficient of ( x ) is ( pq + pr + qr ), which is ( 20 ) when scaled by ( 6 ). Therefore, dividing ( 20 ) by ( 6 ) gives the sum of the products of the roots taken two at a time, which is ( frac{10}{3} ).I think that's solid. I don't see any mistakes in my reasoning. So, the value of ( pq + pr + qr ) is ( frac{10}{3} ).**Final Answer**The value of ( pq + pr + qr ) is boxed{dfrac{10}{3}}.

💡First, I recognize that the operator ( a text{ Y } b ) is defined as ( a^2 - 2ab + b^2 ).To find the value of ( 5 text{ Y } 3 ), I substitute ( a = 5 ) and ( b = 3 ) into the expression:[5 text{ Y } 3 = 5^2 - 2(5)(3) + 3^2]Next, I calculate each term individually:[5^2 = 25][2 times 5 times 3 = 30][3^2 = 9]Now, I substitute these values back into the expression:[5 text{ Y } 3 = 25 - 30 + 9]Finally, I perform the arithmetic operations to find the result:[25 - 30 + 9 = 4]

💡Alright, so I need to convert the complex number (1 + i sqrt{3}) into its exponential form, which is (re^{i theta}). I remember that to do this, I need to find two things: the modulus (r) and the argument (theta).First, let's tackle the modulus (r). The modulus of a complex number (a + ib) is given by the formula:[r = sqrt{a^2 + b^2}]In this case, (a = 1) and (b = sqrt{3}). Plugging these values into the formula:[r = sqrt{1^2 + (sqrt{3})^2} = sqrt{1 + 3} = sqrt{4} = 2]So, the modulus (r) is 2. That part seems straightforward.Now, onto the argument (theta). The argument is the angle that the complex number makes with the positive real axis in the complex plane. To find (theta), I can use the arctangent function because:[theta = arctanleft(frac{b}{a}right)]Here, (a = 1) and (b = sqrt{3}), so:[theta = arctanleft(frac{sqrt{3}}{1}right) = arctan(sqrt{3})]I recall that (arctan(sqrt{3})) corresponds to a standard angle. Thinking back to the unit circle, I know that:[arctan(sqrt{3}) = frac{pi}{3}]This is because (tanleft(frac{pi}{3}right) = sqrt{3}). So, (theta = frac{pi}{3}).Putting it all together, the exponential form of the complex number (1 + i sqrt{3}) is:[re^{i theta} = 2e^{i frac{pi}{3}}]Therefore, the angle (theta) is (frac{pi}{3}).Wait, let me double-check to make sure I didn't make any mistakes. The modulus calculation seems correct: (sqrt{1 + 3} = 2). For the argument, since both the real and imaginary parts are positive, the complex number lies in the first quadrant, which makes sense for an angle of (frac{pi}{3}). Also, (cosleft(frac{pi}{3}right) = frac{1}{2}) and (sinleft(frac{pi}{3}right) = frac{sqrt{3}}{2}), which matches the original complex number when scaled by the modulus (2). So, everything checks out.I think I've got it right. The exponential form is (2e^{i frac{pi}{3}}) and the angle (theta) is (frac{pi}{3}).