answer：First, I know that Ava has 24 books.Since Ava has twice the number of books as Ethan, I can find Ethan's number of books by dividing Ava's books by 2. So, Ethan has 24 / 2 = 12 books.Julian has three times as many books as Ava. Therefore, Julian has 3 * 24 = 72 books.To find the total number of books the three have together, I add up their individual amounts: 24 (Ava) + 12 (Ethan) + 72 (Julian) = 108 books.Finally, to find three times the total number of books, I multiply the total by 3: 3 * 108 = 324.

answer：First, I need to determine how many apple tarts Emma sold. Each apple tart is cut into 6 slices, and there were 72 customers who ordered apple tart slices. To find the number of apple tarts, I divide the total number of apple slices by the number of slices per tart: 72 ÷ 6 = 12 apple tarts.Next, I'll calculate the number of cherry tarts sold. Each cherry tart is cut into 8 slices, and there were 96 customers who ordered cherry tart slices. Dividing the total number of cherry slices by the number of slices per tart gives: 96 ÷ 8 = 12 cherry tarts.Finally, to find the total number of tarts sold, I add the number of apple tarts and cherry tarts together: 12 + 12 = 24 tarts.

answer：First, I need to determine the value of a dime when spent at a shop. A dime is worth 0.10.Next, I'll calculate the value of a dime if it's melted down. Since each dime weighs 1/10 of an ounce and the value per ounce when melted is 50, the value of one dime when melted is 50 multiplied by 1/10, which equals 5.00.Finally, to find out how many times more money Ava would get from melting the dimes instead of spending them, I'll divide the melted value by the spent value: 5.00 divided by 0.10 equals 50. Therefore, Ava would get 50 times more money by melting the dimes.

answer：First, I need to determine the total number of points Emily has earned from her quizzes. Since she scored 25 points on each of the 4 quizzes, I multiply 25 by 4 to get 100 points.Next, I account for the points she lost due to late submissions. She lost 8 points, so I subtract that from her total earned points: 100 minus 8 equals 92 points.Emily wants to maintain an average of at least 80 points. To find out how many more points she can afford to lose, I subtract her current total from the required average: 80 minus 92 equals -12. Since a negative number of points lost doesn't make sense, this means Emily cannot afford to lose any additional points and still qualify for the honor roll.

answer：Okay, so I have this problem about a geometric sequence. All the terms are positive, which is good because that might help later on. The given equation is ( a_5a_6 + a_4a_7 = 18 ). I need to find the value of ( log_{3}a_1 + log_{3}a_2 + ldots + log_{3}a_{10} ).First, let me recall what a geometric sequence is. It's a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio, usually denoted as ( r ). So, in general, the ( n )-th term of a geometric sequence can be written as ( a_n = a_1 cdot r^{n-1} ), where ( a_1 ) is the first term.Given that, let me write expressions for the terms mentioned in the problem.- ( a_4 = a_1 cdot r^{3} )- ( a_5 = a_1 cdot r^{4} )- ( a_6 = a_1 cdot r^{5} )- ( a_7 = a_1 cdot r^{6} )So, substituting these into the given equation ( a_5a_6 + a_4a_7 = 18 ):First, compute ( a_5a_6 ):( a_5a_6 = (a_1 r^4)(a_1 r^5) = a_1^2 r^{9} )Next, compute ( a_4a_7 ):( a_4a_7 = (a_1 r^3)(a_1 r^6) = a_1^2 r^{9} )So, adding them together:( a_5a_6 + a_4a_7 = a_1^2 r^{9} + a_1^2 r^{9} = 2a_1^2 r^{9} = 18 )So, ( 2a_1^2 r^{9} = 18 ). Let me simplify this equation.Divide both sides by 2:( a_1^2 r^{9} = 9 )So, ( a_1^2 r^{9} = 9 ). Hmm, that's one equation, but I don't know ( a_1 ) or ( r ). Maybe I can express this in terms of another variable or find a relationship between ( a_1 ) and ( r ).Wait, the problem is asking for the sum of logarithms: ( log_{3}a_1 + log_{3}a_2 + ldots + log_{3}a_{10} ). Maybe I can express this sum in terms of ( a_1 ) and ( r ), and then use the equation I have to find its value.Let me recall that the sum of logarithms is the logarithm of the product. So,( log_{3}a_1 + log_{3}a_2 + ldots + log_{3}a_{10} = log_{3}(a_1 a_2 ldots a_{10}) )So, I need to compute the product ( a_1 a_2 ldots a_{10} ), and then take the logarithm base 3 of that product.Let me compute the product ( P = a_1 a_2 ldots a_{10} ).Since each term is a geometric sequence, ( a_k = a_1 r^{k-1} ). So, the product is:( P = a_1 cdot a_2 cdot ldots cdot a_{10} = a_1 cdot (a_1 r) cdot (a_1 r^2) cdot ldots cdot (a_1 r^{9}) )So, that's 10 terms, each with ( a_1 ) and powers of ( r ) from 0 to 9.Let me write this as:( P = (a_1)^{10} cdot r^{0 + 1 + 2 + ldots + 9} )The exponent on ( r ) is the sum of the first 10 non-negative integers, which is ( frac{9 times 10}{2} = 45 ). So,( P = (a_1)^{10} cdot r^{45} )Therefore, the product ( P = a_1^{10} r^{45} ).So, the sum of the logarithms is:( log_{3} P = log_{3}(a_1^{10} r^{45}) )Using logarithm properties, this is:( 10 log_{3} a_1 + 45 log_{3} r )Hmm, so I need to find ( 10 log_{3} a_1 + 45 log_{3} r ). Let me see if I can relate this to the equation I had earlier.Earlier, I had ( a_1^2 r^{9} = 9 ). Let me take the logarithm base 3 of both sides:( log_{3}(a_1^2 r^{9}) = log_{3} 9 )Simplify the left side:( 2 log_{3} a_1 + 9 log_{3} r = 2 )Because ( log_{3} 9 = 2 ).So, ( 2 log_{3} a_1 + 9 log_{3} r = 2 ).Hmm, now, in the expression I need to find, which is ( 10 log_{3} a_1 + 45 log_{3} r ), I notice that 10 is 5 times 2, and 45 is 5 times 9. So, if I multiply both sides of the equation by 5, I get:( 10 log_{3} a_1 + 45 log_{3} r = 10 )Therefore, the sum ( log_{3}a_1 + log_{3}a_2 + ldots + log_{3}a_{10} = 10 ).Wait, let me verify that step again. So, if I have ( 2 log a + 9 log b = 2 ), then multiplying both sides by 5 gives ( 10 log a + 45 log b = 10 ). Yes, that seems correct.So, the sum of the logs is 10.Alternatively, let me think if there's another way to approach this, just to make sure.Another approach: Since the product ( P = a_1 a_2 ldots a_{10} = a_1^{10} r^{45} ), and I know from the given equation that ( a_1^2 r^9 = 9 ). So, ( a_1^{10} r^{45} ) is ( (a_1^2 r^9)^5 ). Because ( (a_1^2 r^9)^5 = a_1^{10} r^{45} ). Therefore, ( P = 9^5 ).So, ( P = 9^5 ). Therefore, ( log_{3} P = log_{3} 9^5 = 5 log_{3} 9 = 5 times 2 = 10 ). So, same result.Yes, that's a quicker way. So, since ( a_1^2 r^9 = 9 ), then ( (a_1^2 r^9)^5 = 9^5 ), which is the product ( P ). Hence, ( log_{3} P = 10 ).Therefore, the value is 10.I think that's solid. I considered two approaches: one by manipulating the logarithms directly, and another by expressing the product in terms of the given equation. Both lead to the same conclusion, so I feel confident.**Final Answer**boxed{10}

answer：Okay, so I have this inequality to solve: ((y + sqrt{x})(y - x^2) sqrt{1 - x} leq 0). I need to find the area of the set of points (x, y) that satisfy this inequality. Hmm, let me try to break this down step by step.First, I notice that the expression is a product of three terms: ((y + sqrt{x})), ((y - x^2)), and (sqrt{1 - x}). The inequality is saying that this product is less than or equal to zero. So, the product is either negative or zero. I remember that when a product is negative, an odd number of the factors must be negative, and when it's positive, an even number of factors must be negative. But since we have a product of three terms, which is an odd number, the product will be negative if an odd number of the terms are negative.But before I get into that, I should consider the domain of the expression because of the square roots. The term (sqrt{x}) requires that (x geq 0), and (sqrt{1 - x}) requires that (1 - x geq 0), which means (x leq 1). So, combining these, the domain is (0 leq x leq 1). That’s good to know because it restricts the x-values we need to consider.Next, let me analyze each factor separately.1. (sqrt{1 - x}): As I already noted, this is defined for (x leq 1). Also, since it's a square root, it's always non-negative. So, (sqrt{1 - x} geq 0) for all (x) in the domain. The only time it's zero is when (x = 1), because (sqrt{1 - 1} = 0). So, this factor is non-negative throughout the domain and zero only at (x = 1).2. (y + sqrt{x}): This is a linear expression in y. Let me solve for y: (y = -sqrt{x}). So, this expression is zero when (y = -sqrt{x}). For y-values above this curve, (y + sqrt{x}) is positive, and for y-values below, it's negative.3. (y - x^2): Similarly, this is a quadratic expression in y. Solving for y gives (y = x^2). So, this expression is zero when (y = x^2). For y-values above this parabola, (y - x^2) is positive, and below, it's negative.So, the inequality is the product of these three terms: ((y + sqrt{x})(y - x^2) sqrt{1 - x} leq 0). Since (sqrt{1 - x}) is non-negative, the sign of the entire expression depends on the product of the first two terms: ((y + sqrt{x})(y - x^2)). Because if (sqrt{1 - x}) is positive (which it is for (x < 1)), then the sign of the product is determined by the other two factors. However, when (x = 1), (sqrt{1 - x} = 0), so the entire expression is zero regardless of y. So, at (x = 1), all y-values satisfy the inequality because the expression equals zero.Therefore, for (0 leq x < 1), the inequality reduces to ((y + sqrt{x})(y - x^2) leq 0), because (sqrt{1 - x}) is positive, so multiplying by it doesn't change the inequality sign. At (x = 1), the inequality is automatically satisfied for all y.So, now I can focus on the inequality ((y + sqrt{x})(y - x^2) leq 0) for (0 leq x < 1). Let me analyze this product.The product of two terms is less than or equal to zero when one term is non-positive and the other is non-negative. So, either:1. (y + sqrt{x} leq 0) and (y - x^2 geq 0), or2. (y + sqrt{x} geq 0) and (y - x^2 leq 0).Let me consider each case separately.Case 1: (y + sqrt{x} leq 0) and (y - x^2 geq 0).This implies:(y leq -sqrt{x}) and (y geq x^2).But wait, for (x) in [0,1), (sqrt{x}) is between 0 and 1, so (-sqrt{x}) is between -1 and 0. On the other hand, (x^2) is between 0 and 1. So, (y) has to be both less than or equal to a negative number and greater than or equal to a positive number. But that's impossible because a number can't be both less than or equal to a negative and greater than or equal to a positive. So, this case doesn't yield any solutions.Case 2: (y + sqrt{x} geq 0) and (y - x^2 leq 0).This implies:(y geq -sqrt{x}) and (y leq x^2).So, in this case, y is sandwiched between (-sqrt{x}) and (x^2). Therefore, the region defined by this case is the area between the curves (y = -sqrt{x}) and (y = x^2) for (0 leq x < 1).Additionally, at (x = 1), as I noted earlier, all y-values satisfy the inequality because the expression is zero. So, we need to include the line (x = 1) in our region.Therefore, the set of points satisfying the inequality is the region between (y = -sqrt{x}) and (y = x^2) for (0 leq x leq 1), including the line (x = 1).Wait, but hold on. At (x = 1), the expression is zero regardless of y, so all points on (x = 1) are included. But for (x < 1), the region is between (y = -sqrt{x}) and (y = x^2). So, the area we need is the integral from (x = 0) to (x = 1) of the difference between (x^2) and (-sqrt{x}), which is (x^2 - (-sqrt{x}) = x^2 + sqrt{x}).But before I proceed, I should check if these curves intersect somewhere in the interval [0,1]. If they do, the area might be split into different regions. Let me find if (x^2 = -sqrt{x}) for any x in [0,1].Set (x^2 = -sqrt{x}). Since (x^2) is always non-negative and (-sqrt{x}) is non-positive, the only solution is when both sides are zero. That happens at x = 0. So, the curves only intersect at (0,0). Therefore, for all x in (0,1], (x^2) is above (-sqrt{x}), so the region between them is well-defined.Therefore, the area A can be calculated as the integral from x = 0 to x = 1 of [x^2 - (-sqrt{x})] dx, which is the integral of (x^2 + sqrt{x}) dx from 0 to 1.But wait, actually, since the region is between (y = -sqrt{x}) and (y = x^2), the vertical distance between these two curves is (x^2 - (-sqrt{x}) = x^2 + sqrt{x}). So, integrating this from 0 to 1 gives the area.Let me compute that integral:A = ∫₀¹ (x² + √x) dxFirst, let's integrate term by term.The integral of x² dx is (x³)/3.The integral of √x dx is the integral of x^(1/2) dx, which is (x^(3/2))/(3/2) = (2/3) x^(3/2).So, putting it together:A = [ (x³)/3 + (2/3) x^(3/2) ] from 0 to 1Evaluating at x = 1:(1³)/3 + (2/3)(1)^(3/2) = (1/3) + (2/3)(1) = (1/3) + (2/3) = 1Evaluating at x = 0:(0)/3 + (2/3)(0) = 0So, the area A is 1 - 0 = 1.Wait, but hold on. The integral gave me 1, but I need to make sure I didn't make a mistake. Let me double-check.The integral of x² is indeed (x³)/3, correct. The integral of √x is (2/3) x^(3/2), correct. So, when I plug in x=1, it's (1/3) + (2/3) = 1. At x=0, both terms are zero. So, the area is 1.But let me think again. The region is between (y = x^2) and (y = -sqrt{x}) from x=0 to x=1. So, the vertical distance is (x^2 - (-sqrt{x}) = x^2 + sqrt{x}), which is what I integrated. So, the area is indeed 1.But wait, is that the entire story? Because at x=1, all y-values are included. So, does that affect the area? Hmm, actually, at x=1, the set of points is the entire vertical line x=1, which is a line with zero area. So, it doesn't contribute to the area. Therefore, the total area is just 1.But let me visualize this to make sure. For x between 0 and 1, the region is between the parabola (y = x^2) and the curve (y = -sqrt{x}). So, it's a sort of "band" between these two curves from x=0 to x=1. At x=0, both curves meet at (0,0). As x increases, (y = x^2) goes up and (y = -sqrt{x}) goes down. So, the area between them is indeed the integral of their difference.Wait, but actually, the integral I computed was the area between (y = x^2) and (y = -sqrt{x}), which is correct. So, the area is 1.But just to be thorough, let me consider the inequality again. The original inequality is ((y + sqrt{x})(y - x^2) sqrt{1 - x} leq 0). Since (sqrt{1 - x}) is non-negative, the inequality is equivalent to ((y + sqrt{x})(y - x^2) leq 0) for (x < 1), and all y when (x = 1). So, the region is indeed between (y = -sqrt{x}) and (y = x^2) for (x) in [0,1), and the line x=1.But since the line x=1 has zero area, the total area is just the integral from 0 to 1 of (x² + √x) dx, which is 1.Wait, but hold on. Let me compute the integral again step by step to make sure I didn't make a calculation mistake.Compute ∫₀¹ x² dx:= [x³ / 3] from 0 to 1= (1³ / 3) - (0³ / 3)= 1/3 - 0= 1/3Compute ∫₀¹ √x dx:= ∫₀¹ x^(1/2) dx= [ (x^(3/2) ) / (3/2) ] from 0 to 1= (2/3) [x^(3/2)] from 0 to 1= (2/3)(1^(3/2) - 0^(3/2))= (2/3)(1 - 0)= 2/3So, adding both integrals:1/3 + 2/3 = 1Yes, that's correct. So, the area is indeed 1.But wait, just to make sure, let me think about the region again. The region is bounded between two curves from x=0 to x=1. At x=0, both curves meet at (0,0). As x increases, (y = x^2) goes up and (y = -sqrt{x}) goes down. So, the area between them is the area under (y = x^2) minus the area under (y = -sqrt{x}). But since (y = -sqrt{x}) is below the x-axis, the area between them is actually the area under (y = x^2) plus the area above (y = -sqrt{x}) (which is the same as the integral of |y| for (y = -sqrt{x})).Wait, actually, when computing the area between two curves, it's the integral of the upper function minus the lower function. In this case, (y = x^2) is above (y = -sqrt{x}) for all x in (0,1]. So, the vertical distance is (x^2 - (-sqrt{x}) = x^2 + sqrt{x}). Therefore, integrating this from 0 to 1 gives the total area between the curves, which is 1.So, I think I'm confident that the area is 1.But just to make sure, let me consider a small x, say x=0.25. Then, (y = x^2 = 0.0625) and (y = -sqrt{x} = -0.5). So, the vertical distance is 0.0625 - (-0.5) = 0.5625, which is 0.25² + sqrt(0.25) = 0.0625 + 0.5 = 0.5625. So, that checks out.Another point, x=0.5: (y = x^2 = 0.25), (y = -sqrt{x} ≈ -0.7071). The vertical distance is 0.25 - (-0.7071) ≈ 0.9571, which is 0.5² + sqrt(0.5) ≈ 0.25 + 0.7071 ≈ 0.9571. Correct.So, the integrand is correct, and the integral evaluates to 1. Therefore, the area is 1.Wait, but hold on. Let me think about the inequality again. The original inequality is ((y + sqrt{x})(y - x^2) sqrt{1 - x} leq 0). So, for (x < 1), since (sqrt{1 - x}) is positive, the inequality reduces to ((y + sqrt{x})(y - x^2) leq 0). So, the region is where the product is negative or zero, which is between the two curves (y = -sqrt{x}) and (y = x^2). So, that's correct.But wait, another way to think about it is that the product is negative when one factor is positive and the other is negative. So, either:1. (y + sqrt{x} geq 0) and (y - x^2 leq 0), which is (y geq -sqrt{x}) and (y leq x^2), which is the region between the curves.Or2. (y + sqrt{x} leq 0) and (y - x^2 geq 0), which would be (y leq -sqrt{x}) and (y geq x^2). But as I thought earlier, this is impossible because (x^2) is non-negative and (-sqrt{x}) is non-positive, so there's no overlap except at x=0, y=0.So, only the first case contributes to the region, which is between (y = -sqrt{x}) and (y = x^2).Therefore, the area is indeed 1.But just to make sure, let me compute the integral once more:A = ∫₀¹ (x² + √x) dx= ∫₀¹ x² dx + ∫₀¹ √x dx= [x³ / 3]₀¹ + [ (2/3) x^(3/2) ]₀¹= (1/3 - 0) + (2/3 - 0)= 1/3 + 2/3= 1Yes, that's correct.So, after all this thinking and checking, I believe the area is 1.**Final Answer**The area of the set of points is boxed{1}.