💡Okay, so I have this problem here where R is equal to 3 raised to the power of a, and S is equal to 5 raised to the power of b. I need to figure out which of the given options is equal to 15 raised to the power of ab for any integers a and b. The options are:A) R^b S^aB) R^{2b} S^aC) R^a S^{2b}D) R^{2a} S^bE) R^{3a} S^bAlright, let me break this down step by step. First, I know that 15 is 3 multiplied by 5. So, 15^{ab} can be written as (3*5)^{ab}. Using the properties of exponents, I can separate this into 3^{ab} multiplied by 5^{ab}. Now, since R is 3^a, if I raise R to the power of b, that would be (3^a)^b, which simplifies to 3^{ab}. Similarly, S is 5^b, so if I raise S to the power of a, that would be (5^b)^a, which simplifies to 5^{ab}. So, putting these together, 3^{ab} times 5^{ab} is exactly R^b times S^a. That means 15^{ab} is equal to R^b S^a, which is option A. Let me double-check to make sure I didn't make a mistake. If I substitute R and S back into option A, I get (3^a)^b times (5^b)^a, which is 3^{ab} times 5^{ab}, and that's the same as (3*5)^{ab}, which is 15^{ab}. Yep, that seems right. I guess the other options don't work because they either have the exponents in the wrong place or they have extra factors. For example, option B has R^{2b} which would be (3^a)^{2b} = 3^{2ab}, which is more than what we need. Similarly, option C has S^{2b}, which would be (5^b)^{2a} = 5^{2ab}, again too much. Options D and E have similar issues with their exponents. So, option A is the only one that correctly gives us 15^{ab}.I think I'm confident now that option A is the correct answer.**Final Answer**boxed{A}

💡Okay, so I have this problem here: Given that the magnitude of vector a is 3, the magnitude of vector b is 5, and vector a is parallel to vector b, I need to find the dot product of a and b. Hmm, let's see how to approach this.First, I remember that the dot product of two vectors can be calculated using the formula:[vec{a} cdot vec{b} = |vec{a}| |vec{b}| cos theta]where ( theta ) is the angle between the two vectors. Since the problem states that ( vec{a} parallel vec{b} ), that means the angle between them is either 0 degrees or 180 degrees. Wait, why both 0 and 180 degrees? Oh, right! If two vectors are parallel, they can be pointing in the same direction (0 degrees) or in exactly opposite directions (180 degrees). So, I need to consider both possibilities.Let me write down the given values:- ( |vec{a}| = 3 )- ( |vec{b}| = 5 )Now, plugging these into the dot product formula for both angles.First, if ( theta = 0^circ ):[vec{a} cdot vec{b} = 3 times 5 times cos 0^circ]I know that ( cos 0^circ = 1 ), so this simplifies to:[vec{a} cdot vec{b} = 15 times 1 = 15]Okay, that's straightforward. Now, what if ( theta = 180^circ )?[vec{a} cdot vec{b} = 3 times 5 times cos 180^circ]And ( cos 180^circ = -1 ), so:[vec{a} cdot vec{b} = 15 times (-1) = -15]So, depending on whether the vectors are pointing in the same direction or opposite directions, the dot product can be either 15 or -15.Wait, does the problem specify anything else about the vectors? It just says they are parallel. So, both cases are possible. Therefore, the dot product can be either positive or negative 15.Let me double-check if I missed anything. The magnitudes are given, and the only condition is that they are parallel. So, the angle is either 0 or 180 degrees, leading to the dot product being either 15 or -15. Yeah, that seems right.I think I've got it. The dot product is either 15 or -15 when the vectors are parallel. So, the answer should reflect both possibilities.

💡Alright, so I have this problem about an odd function, and I need to figure out two things: first, determine if the function is increasing or decreasing and prove it; second, solve an inequality involving this function. Let me take it step by step.Starting with part (1). The function is given as f(x) = a/(e^x + 1) + 1, and it's an odd function. I remember that an odd function satisfies the property f(-x) = -f(x) for all x in its domain. So, I can use this property to find the value of 'a' first, right?Let me write down the equation:f(-x) = -f(x)Substituting the given function:a/(e^{-x} + 1) + 1 = -[a/(e^x + 1) + 1]Simplify the left side. Since e^{-x} is 1/e^x, so:a/(1/e^x + 1) + 1 = a/( (1 + e^x)/e^x ) + 1 = a*e^x/(1 + e^x) + 1So, left side becomes a*e^x/(e^x + 1) + 1.Right side is -a/(e^x + 1) - 1.So, setting them equal:a*e^x/(e^x + 1) + 1 = -a/(e^x + 1) - 1Let me bring all terms to one side:a*e^x/(e^x + 1) + 1 + a/(e^x + 1) + 1 = 0Combine like terms:[a*e^x + a]/(e^x + 1) + 2 = 0Factor out 'a' in the numerator:a*(e^x + 1)/(e^x + 1) + 2 = 0Simplify:a + 2 = 0So, a = -2.Okay, so now we know that a is -2. So, the function becomes:f(x) = (-2)/(e^x + 1) + 1Simplify that:f(x) = 1 - 2/(e^x + 1)Now, to determine the monotonicity, I need to find the derivative of f(x). Let's compute f'(x).f(x) = 1 - 2/(e^x + 1)So, f'(x) = derivative of 1 is 0, minus derivative of 2/(e^x + 1).Using the quotient rule: derivative of 2/(e^x + 1) is -2*(e^x)/(e^x + 1)^2.So, f'(x) = - [ -2*e^x/(e^x + 1)^2 ] = 2*e^x/(e^x + 1)^2.Since e^x is always positive, and (e^x + 1)^2 is also always positive, the derivative f'(x) is positive for all x. Therefore, f(x) is a monotonically increasing function.Alright, that takes care of part (1). Now, moving on to part (2). I need to solve the inequality:f(log_2^2 x) + f(log_{sqrt{2}} x - 3) ≤ 0First, let me parse the notation. log_2^2 x probably means (log_2 x)^2, right? So, it's the square of the logarithm base 2 of x. Similarly, log_{sqrt{2}} x is the logarithm base sqrt(2) of x.So, let me rewrite the inequality with this understanding:f( (log_2 x)^2 ) + f( log_{sqrt{2}} x - 3 ) ≤ 0Since f is an odd function, I know that f(-y) = -f(y). Maybe I can use this property to simplify the inequality.Let me denote:A = (log_2 x)^2B = log_{sqrt{2}} x - 3So, the inequality becomes:f(A) + f(B) ≤ 0I need to express this in terms that can be simplified. Since f is odd, maybe I can relate f(B) to -f(-B). Let's see.But first, perhaps it's better to express log_{sqrt{2}} x in terms of log base 2. Because sqrt(2) is 2^{1/2}, so log_{sqrt{2}} x is equal to log_{2^{1/2}} x, which is equal to 2*log_2 x. Because log_{a^b} c = (1/b) log_a c. Wait, actually, log_{2^{1/2}} x = (log_2 x)/(log_2 2^{1/2}) = (log_2 x)/(1/2) = 2 log_2 x.So, log_{sqrt{2}} x = 2 log_2 x.Therefore, B = 2 log_2 x - 3.So, now, the inequality becomes:f( (log_2 x)^2 ) + f( 2 log_2 x - 3 ) ≤ 0Let me denote t = log_2 x. Then, (log_2 x)^2 = t^2, and 2 log_2 x - 3 = 2t - 3.So, substituting, the inequality becomes:f(t^2) + f(2t - 3) ≤ 0Now, since f is an odd function, f(-y) = -f(y). Let me see if I can express one of these terms as f(-something).But first, let me recall that f is monotonically increasing, as we found in part (1). So, f is increasing, which might help in solving the inequality.Let me write the inequality again:f(t^2) + f(2t - 3) ≤ 0Since f is odd, f(-y) = -f(y). So, if I can write f(2t - 3) as -f(something), then maybe I can combine the terms.But let me think differently. Since f is odd, f(2t - 3) = -f(-(2t - 3)) = -f(3 - 2t). So, substituting back:f(t^2) - f(3 - 2t) ≤ 0So, f(t^2) ≤ f(3 - 2t)Since f is monotonically increasing, if f(a) ≤ f(b), then a ≤ b.Therefore, t^2 ≤ 3 - 2tSo, the inequality reduces to:t^2 + 2t - 3 ≤ 0Let me solve this quadratic inequality.First, find the roots of t^2 + 2t - 3 = 0.Using quadratic formula:t = [-2 ± sqrt(4 + 12)] / 2 = [-2 ± sqrt(16)] / 2 = [-2 ± 4]/2So, t = (-2 + 4)/2 = 2/2 = 1t = (-2 - 4)/2 = -6/2 = -3So, the roots are t = 1 and t = -3.The quadratic t^2 + 2t - 3 factors as (t + 3)(t - 1). Since the coefficient of t^2 is positive, the parabola opens upwards, so the inequality t^2 + 2t - 3 ≤ 0 holds between the roots.Therefore, t ∈ [-3, 1]So, t is between -3 and 1.But t = log_2 x, so:-3 ≤ log_2 x ≤ 1Convert this back to x.Recall that log_2 x = y implies x = 2^y.So, for log_2 x ≥ -3: x ≥ 2^{-3} = 1/8And for log_2 x ≤ 1: x ≤ 2^1 = 2Therefore, x ∈ [1/8, 2]But wait, I need to make sure that the arguments of the logarithms in the original inequality are defined.In the original inequality, we have log_2 x and log_{sqrt{2}} x. Both logarithms are defined when x > 0. So, x must be positive.Therefore, the solution is x ∈ [1/8, 2]Let me double-check if I missed anything.First, in part (1), I found a = -2, so f(x) = 1 - 2/(e^x + 1). Then, I computed the derivative and found it positive, so f is increasing. That seems correct.In part (2), I converted log_{sqrt{2}} x to 2 log_2 x, which is correct because log_{2^{1/2}} x = 2 log_2 x.Then, I set t = log_2 x, so the inequality became f(t^2) + f(2t - 3) ≤ 0. Then, using f is odd, I rewrote f(2t - 3) as -f(3 - 2t), leading to f(t^2) ≤ f(3 - 2t). Since f is increasing, this implies t^2 ≤ 3 - 2t, which simplifies to t^2 + 2t - 3 ≤ 0, solved to t ∈ [-3, 1], which translates to x ∈ [1/8, 2].I think that's correct. I don't see any mistakes in the steps. So, the solution should be x between 1/8 and 2, inclusive.

💡Okay, so I have this sequence of numbers: -1/2, 1/2, -3/8, 1/4, ..., -5/32, 3/32, ... and I need to find the 8th number in the sequence. Hmm, let's see. First, I should try to understand the pattern here.Looking at the numbers, they alternate between negative and positive. The first term is negative, the second is positive, the third is negative, and so on. So, it seems like the sign alternates with each term. That's one pattern.Now, let's look at the fractions themselves. The first term is -1/2, the second is 1/2, the third is -3/8, the fourth is 1/4, and then it continues with -5/32 and 3/32. I notice that the denominators are powers of 2: 2, 4, 8, 16, 32, etc. So, the denominator of the nth term is 2^n.What about the numerators? The first term has a numerator of 1, the second also has 1, the third has 3, the fourth has 1, the fifth has 5, and the sixth has 3. Hmm, that seems a bit more complicated. It looks like the numerators are increasing by 2 every two terms. Let me write them out:Term 1: -1/2 (numerator 1)Term 2: 1/2 (numerator 1)Term 3: -3/8 (numerator 3)Term 4: 1/4 (numerator 1)Term 5: -5/32 (numerator 5)Term 6: 3/32 (numerator 3)Term 7: ?Term 8: ?Wait, so the numerators are 1, 1, 3, 1, 5, 3,... It seems like for odd terms, the numerator increases by 2 each time, starting from 1: 1, 3, 5,... And for even terms, the numerator is 1, then 3, then maybe 5? But in the given sequence, the even terms are 1/2, 1/4, 3/32. Hmm, that doesn't quite fit.Let me think differently. Maybe the numerator is related to the term's position. Let's see:Term 1: numerator 1Term 2: numerator 1Term 3: numerator 3Term 4: numerator 1Term 5: numerator 5Term 6: numerator 3Term 7: ?Term 8: ?It seems like for odd terms, the numerator is the same as the term number divided by 2, rounded up or something. Wait, term 1: 1, term 3: 3, term 5:5. So, for odd terms, numerator is (term number +1)/2. Let's check:Term 1: (1+1)/2=1, correct.Term 3: (3+1)/2=2, but numerator is 3. Hmm, not quite.Wait, maybe it's just the term number divided by 2, rounded up. Term 1:1, term3:2, but numerator is 3. Doesn't fit.Alternatively, maybe the numerator for odd terms is the term number itself. Term1:1, term3:3, term5:5. That makes sense. So, for odd terms, numerator is the term number, and for even terms, numerator is 1,3,5,... but starting from term2.Wait, term2: numerator1, term4:numerator1, term6:numerator3. Hmm, not consistent.Wait, term2 is 1/2, term4 is 1/4, term6 is3/32. So, term2 and term4 have numerator1, term6 has numerator3. Maybe for even terms, the numerator is (term number)/2 -1. Let's see:Term2: (2)/2 -1=0, but numerator is1. Not matching.Alternatively, maybe the numerator for even terms is (term number -2)/2. Term2:0, term4:1, term6:2. But term2 numerator is1, term4 is1, term6 is3. Doesn't fit.Wait, maybe the numerator for even terms is following a different pattern. Term2:1, term4:1, term6:3. Maybe it's increasing by 0, then 2. Not sure.Alternatively, perhaps the numerators are following a pattern where odd terms have numerators increasing by 2, starting from1, and even terms have numerators increasing by 2 starting from1 as well, but shifted.Wait, term1:1, term3:3, term5:5, term7:7,... So, odd terms have numerators 1,3,5,7,...Term2:1, term4:1, term6:3, term8:?Hmm, term2 and term4 have numerator1, term6 has numerator3. Maybe for even terms, the numerator is (term number)/2 - something.Term2:1, term4:1, term6:3. If term number is 2,4,6,... then term number divided by 2 is1,2,3,... So, maybe numerator for even terms is term number divided by2 minus0, but term2:1=1, term4:1=2-1, term6:3=3. Hmm, not consistent.Wait, term2:1, term4:1, term6:3. Maybe it's like floor((term number)/4)*2 +1. Term2: floor(2/4)=0, 0*2+1=1. Term4: floor(4/4)=1, 1*2+1=3. But term4 has numerator1, not3. Doesn't fit.This is getting complicated. Maybe I should look at the sequence differently. Let's write out the terms with their positions:Term1: -1/2Term2: 1/2Term3: -3/8Term4: 1/4Term5: -5/32Term6: 3/32Term7: ?Term8: ?Looking at the denominators: 2,2,8,4,32,32,...Wait, denominators: 2,2,8,4,32,32,...Hmm, not sure. Maybe denominators are 2^n where n is the term number.Term1:2^1=2Term2:2^2=4, but denominator is2. Doesn't fit.Wait, term1:2, term2:2, term3:8, term4:4, term5:32, term6:32.Wait, term1:2=2^1term2:2=2^1term3:8=2^3term4:4=2^2term5:32=2^5term6:32=2^5Hmm, so denominators are 2^{n} where n is the term number for odd terms, and for even terms, it's 2^{n/2}.Wait, term1:2^1=2term2:2^{2/2}=2^1=2term3:2^3=8term4:2^{4/2}=2^2=4term5:2^5=32term6:2^{6/2}=2^3=8, but term6 has denominator32. Hmm, doesn't fit.Wait, term6 has denominator32, which is2^5, but 6/2=3, 2^3=8. So, that doesn't match.Maybe denominators are 2^{ceil(n/2)}.Term1:ceil(1/2)=1, 2^1=2Term2:ceil(2/2)=1, 2^1=2Term3:ceil(3/2)=2, 2^2=4, but term3 has denominator8. Doesn't fit.Alternatively, denominators are 2^{floor(n/2)+1}.Term1:floor(1/2)+1=0+1=1, 2^1=2Term2:floor(2/2)+1=1+1=2, 2^2=4, but term2 has denominator2. Doesn't fit.This approach isn't working. Maybe I should look at the numerators and denominators separately.Numerators: -1,1,-3,1,-5,3,...Denominators:2,2,8,4,32,32,...Wait, denominators:2,2,8,4,32,32,...Looking at denominators, it seems like for odd terms, the denominator is 2^{term number}, and for even terms, it's 2^{term number/2}.Term1:2^1=2Term2:2^{2/2}=2^1=2Term3:2^3=8Term4:2^{4/2}=2^2=4Term5:2^5=32Term6:2^{6/2}=2^3=8, but term6 has denominator32. Hmm, not matching.Wait, term6 has denominator32, which is2^5, but 6/2=3, 2^3=8. So, that doesn't fit.Maybe denominators are 2^{n} for all terms, but that doesn't fit term2 and term4.Wait, term1:2^1=2term2:2^2=4, but denominator is2. Doesn't fit.term3:2^3=8term4:2^4=16, but denominator is4. Doesn't fit.term5:2^5=32term6:2^6=64, but denominator is32. Doesn't fit.So, denominators are not simply 2^n.Wait, let's see the denominators again:2,2,8,4,32,32,...Looking at the pattern: 2,2,8,4,32,32,...It seems like denominators are doubling every two terms, but not exactly.From term1 to term2:2 to2, same.Term2 to term3:2 to8, multiplied by4.Term3 to term4:8 to4, divided by2.Term4 to term5:4 to32, multiplied by8.Term5 to term6:32 to32, same.Hmm, not a clear pattern.Wait, maybe denominators are 2^{n} for odd terms and 2^{n/2} for even terms.Term1:2^1=2Term2:2^{2/2}=2^1=2Term3:2^3=8Term4:2^{4/2}=2^2=4Term5:2^5=32Term6:2^{6/2}=2^3=8, but term6 has denominator32. Doesn't fit.Wait, term6 has denominator32, which is2^5, but according to this, it should be8. So, maybe it's 2^{ceil(n/2)}.Term1:ceil(1/2)=1, 2^1=2Term2:ceil(2/2)=1, 2^1=2Term3:ceil(3/2)=2, 2^2=4, but term3 has denominator8. Doesn't fit.Alternatively, denominators are 2^{n} for odd terms and 2^{n-1} for even terms.Term1:2^1=2Term2:2^{2-1}=2^1=2Term3:2^3=8Term4:2^{4-1}=2^3=8, but term4 has denominator4. Doesn't fit.Hmm, not working.Maybe I should look at the numerators and denominators together.Term1: -1/2Term2:1/2Term3:-3/8Term4:1/4Term5:-5/32Term6:3/32Term7:?Term8:?Looking at the numerators: -1,1,-3,1,-5,3,...It seems like for odd terms, the numerator is negative and increasing by2: -1,-3,-5,...For even terms, the numerator is positive and seems to be 1,1,3,...Wait, term2:1, term4:1, term6:3. So, for even terms, the numerator starts at1 and increases by2 every two terms.Term2:1Term4:1Term6:3Term8:?So, term8 would have numerator5.Similarly, for odd terms:Term1:-1Term3:-3Term5:-5Term7:-7So, the pattern for numerators is:Odd terms: -1, -3, -5, -7,...Even terms:1,1,3,5,...Now, denominators:Term1:2Term2:2Term3:8Term4:4Term5:32Term6:32Term7:?Term8:?Looking at denominators:Term1:2=2^1Term2:2=2^1Term3:8=2^3Term4:4=2^2Term5:32=2^5Term6:32=2^5Term7:?Term8:?So, denominators for odd terms:2^1,2^3,2^5,...Denominators for even terms:2^1,2^2,2^5,...Wait, that doesn't seem consistent.Wait, term1:2^1term3:2^3term5:2^5So, for odd terms, denominator is2^{term number}.Term1:2^1=2Term3:2^3=8Term5:2^5=32Term7:2^7=128For even terms:Term2:2^1=2Term4:2^2=4Term6:2^5=32Term8:?Wait, term6 denominator is32=2^5, but term6 is even term number6. So, 2^{6/2 + something}.Term2:2^{1}=2Term4:2^{2}=4Term6:2^{5}=32Term8:?Hmm, the exponent for even terms seems to be increasing by3 from term4 to term6: term4 exponent2, term6 exponent5. So, 2,5, which is an increase of3. Maybe term8 exponent would be5+3=8.So, term8 denominator would be2^8=256.But let's check:Term2:2^1=2Term4:2^2=4Term6:2^5=32Term8:2^8=256So, the exponents for even terms are1,2,5,8,...The differences between exponents:1 to2 is1, 2 to5 is3, 5 to8 is3. So, maybe the pattern is +1, +3, +3,...Not sure, but let's assume that term8 denominator is2^8=256.So, putting it all together:Term8 is an even term, so numerator is5 (from the even term numerator pattern:1,1,3,5,...), and denominator is256.So, term8=5/256.But wait, let's check the given terms:Term6:3/32, which is3/32=3/2^5.So, term6 numerator is3, denominator is2^5.Similarly, term8 would be5/2^8=5/256.But let's see if this fits with the previous terms.Term2:1/2=1/2^1Term4:1/4=1/2^2Term6:3/32=3/2^5Term8:5/256=5/2^8Hmm, the exponents for even terms are1,2,5,8,...The exponents are increasing by1, then3, then3. Not a clear pattern, but maybe it's related to the term number.Wait, term2: exponent1=2-1term4: exponent2=4-2term6: exponent5=6-1term8: exponent8=8-0Hmm, not sure.Alternatively, maybe the exponent for even term n is (n/2)^2 - something.Term2: n=2, (2/2)^2=1, exponent1Term4: n=4, (4/2)^2=4, exponent2Term6: n=6, (6/2)^2=9, exponent5Term8: n=8, (8/2)^2=16, exponent8Hmm, not matching.Wait, term6 exponent is5, which is less than9.Maybe exponent for even term n is (n/2) + something.Term2:1=1+0Term4:2=2+0Term6:5=3+2Term8:8=4+4Hmm, not a clear pattern.Alternatively, maybe the exponent for even term n is the sum of previous exponents.Term2:1Term4:1+1=2Term6:2+3=5Term8:5+3=8Wait, that could be a possibility. So, exponents for even terms:1,2,5,8,...Where each exponent is the previous exponent plus the difference between the previous two exponents.From1 to2: difference1From2 to5: difference3From5 to8: difference3So, the differences are1,3,3,...Not sure if that's a valid pattern, but if we follow that, term8 exponent would be8.So, term8 denominator is2^8=256.Therefore, term8=5/256.But let's check if this makes sense with the given terms.Term6:3/32=3/2^5Term8:5/256=5/2^8Yes, that seems consistent with the pattern.So, the 8th term is5/256.But wait, let's see if there's another way to look at it.Maybe the general formula for the nth term is (-1)^n * (n)/2^n.Let's test this:Term1: (-1)^1 *1/2^1=-1/2Term2: (-1)^2 *2/2^2=1*2/4=1/2Term3: (-1)^3 *3/2^3=-3/8Term4: (-1)^4 *4/2^4=4/16=1/4Term5: (-1)^5 *5/2^5=-5/32Term6: (-1)^6 *6/2^6=6/64=3/32Term7: (-1)^7 *7/2^7=-7/128Term8: (-1)^8 *8/2^8=8/256=1/32Wait, that's different. According to this formula, term8 is1/32.But earlier, I thought it was5/256.Which one is correct?Looking back at the given sequence:-1/2,1/2,-3/8,1/4,-5/32,3/32,...So, term7 is missing, but term6 is3/32.If we follow the general formula (-1)^n *n/2^n, then term7 would be-7/128, and term8 would be8/256=1/32.But in the given sequence, term6 is3/32, which matches the formula: term6=6/64=3/32.Similarly, term5=-5/32, which matches.Term4=4/16=1/4, matches.Term3=-3/8, matches.Term2=2/4=1/2, matches.Term1=-1/2, matches.So, this formula seems to fit all the given terms.Therefore, the general formula is a_n=(-1)^n *n/2^n.So, term8=a8=(-1)^8 *8/2^8=1*8/256=1/32.Therefore, the 8th term is1/32.Wait, but earlier I thought the numerator for even terms was increasing by2, but according to this formula, the numerator is just the term number.So, maybe my initial approach was overcomplicating it.The correct pattern is that the nth term is (-1)^n multiplied by n over2^n.Therefore, the 8th term is1/32.

💡To find the maximum value of the quadratic expression (-9z^2 + 27z + 3), I first recognize that it is a quadratic function in the form (az^2 + bz + c), where (a = -9), (b = 27), and (c = 3).Since the coefficient of (z^2) is negative ((a = -9)), the parabola opens downward, indicating that the function has a maximum point at its vertex.The vertex of a quadratic function (az^2 + bz + c) occurs at (z = -frac{b}{2a}). Plugging in the values, I calculate the (z)-coordinate of the vertex:[z = -frac{27}{2 times (-9)} = frac{27}{18} = frac{3}{2}]Next, I substitute (z = frac{3}{2}) back into the original expression to find the maximum value:[-9left(frac{3}{2}right)^2 + 27left(frac{3}{2}right) + 3]Calculating each term:[-9 times frac{9}{4} = -frac{81}{4}][27 times frac{3}{2} = frac{81}{2}][3 = frac{12}{4}]Combining these:[-frac{81}{4} + frac{81}{2} + frac{12}{4} = -frac{81}{4} + frac{162}{4} + frac{12}{4} = frac{93}{4}]Therefore, the maximum value of the quadratic expression is (frac{93}{4}).

💡First, calculate the total miles Tom drives each week:- **Monday, Wednesday, Friday, and Sunday:** 4 days * 50 miles/day = 200 miles- **Tuesday and Thursday:** 2 days * 80 miles/day = 160 miles- **Saturday:** 120 miles**Total miles per week:** 200 + 160 + 120 = 480 milesNext, determine the weekly cost for the miles driven:- **Cost per mile:** 0.1- **Weekly miles:** 480 miles- **Cost for miles per week:** 480 * 0.1 = 48Add the weekly fee:- **Weekly fee:** 100- **Total weekly cost without discount:** 48 + 100 = 148Calculate the discount for every 5th week:- **Discount per 5th week:** 30- **Number of 5th weeks in a year:** 52 / 5 = 10.4 (rounded down to 10)- **Total discount in a year:** 10 * 30 = 300Compute the total cost for the year without monthly tolls and parking fees:- **Total weekly cost without discount * 52 weeks:** 148 * 52 = 7,696- **Subtract total discount:** 7,696 - 300 = 7,396Add the monthly tolls and parking fees:- **Monthly tolls and parking fees:** 50- **Number of months in a year:** 12- **Total tolls and parking fees for the year:** 50 * 12 = 600Finally, calculate the total cost for the year:- **Total cost for the year:** 7,396 + 600 = 7,996Tom has to pay boxed{7,996} in a year for leasing the car.