💡First, I'll define the unknown number as ( x ).According to the problem, the number exceeds its ( frac{3}{8} ) part by 15. This can be written as the equation:[x = frac{3}{8}x + 15]To solve for ( x ), I'll subtract ( frac{3}{8}x ) from both sides to get all the ( x ) terms on one side:[x - frac{3}{8}x = 15]Next, I'll combine the ( x ) terms by finding a common denominator:[frac{8}{8}x - frac{3}{8}x = 15][frac{5}{8}x = 15]To isolate ( x ), I'll divide both sides by ( frac{5}{8} ), which is the same as multiplying by the reciprocal:[x = 15 times frac{8}{5}][x = frac{120}{5}][x = 24]Therefore, the number is 24.

💡Alright, so I have this problem about a hyperbola, and I need to find its eccentricity. Let me try to understand what's given and what I need to find.The hyperbola is given by the equation ( dfrac{x^2}{a^2} - dfrac{y^2}{b^2} = 1 ), where ( a > 0 ) and ( b > 0 ). The right focus of this hyperbola is denoted by ( F ). There's a point ( P ) on the hyperbola such that the triangle formed by the origin ( O ), the focus ( F ), and the point ( P ) is an equilateral triangle. I need to find the eccentricity ( e ) of the hyperbola.First, let me recall some properties of hyperbolas. The standard form is ( dfrac{x^2}{a^2} - dfrac{y^2}{b^2} = 1 ). The foci of a hyperbola are located at ( (pm c, 0) ), where ( c^2 = a^2 + b^2 ). The eccentricity ( e ) is defined as ( e = dfrac{c}{a} ), and since ( c > a ) for hyperbolas, ( e > 1 ).Given that ( F ) is the right focus, its coordinates are ( (c, 0) ). The origin ( O ) is at ( (0, 0) ). The point ( P ) is somewhere on the hyperbola, and triangle ( OPF ) is equilateral. So, all sides of this triangle are equal in length. That means ( OP = PF = OF ).Since ( OF ) is the distance from the origin to the focus, which is ( c ), so ( OP = PF = c ). Therefore, the distance from ( O ) to ( P ) is ( c ), and the distance from ( P ) to ( F ) is also ( c ).Let me denote the coordinates of point ( P ) as ( (x, y) ). Since ( P ) lies on the hyperbola, it must satisfy the equation ( dfrac{x^2}{a^2} - dfrac{y^2}{b^2} = 1 ).Now, the distance from ( O ) to ( P ) is ( sqrt{x^2 + y^2} = c ). So, squaring both sides, we get:[ x^2 + y^2 = c^2 quad (1) ]Similarly, the distance from ( P ) to ( F ) is ( sqrt{(x - c)^2 + y^2} = c ). Squaring both sides:[ (x - c)^2 + y^2 = c^2 quad (2) ]Let me subtract equation (1) from equation (2) to eliminate ( y^2 ):[ (x - c)^2 + y^2 - (x^2 + y^2) = c^2 - c^2 ][ (x^2 - 2cx + c^2) + y^2 - x^2 - y^2 = 0 ]Simplify:[ -2cx + c^2 = 0 ][ -2cx + c^2 = 0 ]Divide both sides by ( c ) (since ( c neq 0 )):[ -2x + c = 0 ][ -2x + c = 0 ][ 2x = c ][ x = dfrac{c}{2} ]So, the x-coordinate of point ( P ) is ( dfrac{c}{2} ). Now, let's substitute this back into equation (1) to find ( y ):[ left( dfrac{c}{2} right)^2 + y^2 = c^2 ][ dfrac{c^2}{4} + y^2 = c^2 ]Subtract ( dfrac{c^2}{4} ) from both sides:[ y^2 = c^2 - dfrac{c^2}{4} ][ y^2 = dfrac{3c^2}{4} ][ y = pm dfrac{sqrt{3}c}{2} ]So, the coordinates of point ( P ) are ( left( dfrac{c}{2}, dfrac{sqrt{3}c}{2} right) ) or ( left( dfrac{c}{2}, -dfrac{sqrt{3}c}{2} right) ). Since the hyperbola is symmetric, both points will satisfy the hyperbola equation.Now, let's substitute ( x = dfrac{c}{2} ) and ( y = dfrac{sqrt{3}c}{2} ) into the hyperbola equation:[ dfrac{left( dfrac{c}{2} right)^2}{a^2} - dfrac{left( dfrac{sqrt{3}c}{2} right)^2}{b^2} = 1 ]Simplify each term:[ dfrac{dfrac{c^2}{4}}{a^2} - dfrac{dfrac{3c^2}{4}}{b^2} = 1 ][ dfrac{c^2}{4a^2} - dfrac{3c^2}{4b^2} = 1 ]Let me factor out ( dfrac{c^2}{4} ):[ dfrac{c^2}{4} left( dfrac{1}{a^2} - dfrac{3}{b^2} right) = 1 ]Now, I know that for hyperbolas, ( c^2 = a^2 + b^2 ). So, let me express ( b^2 ) in terms of ( a^2 ) and ( c^2 ):[ b^2 = c^2 - a^2 ]Substitute ( b^2 ) into the equation:[ dfrac{c^2}{4} left( dfrac{1}{a^2} - dfrac{3}{c^2 - a^2} right) = 1 ]Let me compute the expression inside the parentheses:[ dfrac{1}{a^2} - dfrac{3}{c^2 - a^2} ]To combine these fractions, find a common denominator, which is ( a^2(c^2 - a^2) ):[ dfrac{c^2 - a^2 - 3a^2}{a^2(c^2 - a^2)} ]Simplify the numerator:[ c^2 - a^2 - 3a^2 = c^2 - 4a^2 ]So, the expression becomes:[ dfrac{c^2 - 4a^2}{a^2(c^2 - a^2)} ]Now, substitute back into the main equation:[ dfrac{c^2}{4} times dfrac{c^2 - 4a^2}{a^2(c^2 - a^2)} = 1 ]Simplify:[ dfrac{c^2(c^2 - 4a^2)}{4a^2(c^2 - a^2)} = 1 ]Let me write this as:[ dfrac{c^2(c^2 - 4a^2)}{4a^2(c^2 - a^2)} = 1 ]Multiply both sides by ( 4a^2(c^2 - a^2) ):[ c^2(c^2 - 4a^2) = 4a^2(c^2 - a^2) ]Expand both sides:Left side:[ c^4 - 4a^2c^2 ]Right side:[ 4a^2c^2 - 4a^4 ]So, the equation becomes:[ c^4 - 4a^2c^2 = 4a^2c^2 - 4a^4 ]Bring all terms to the left side:[ c^4 - 4a^2c^2 - 4a^2c^2 + 4a^4 = 0 ]Combine like terms:[ c^4 - 8a^2c^2 + 4a^4 = 0 ]This is a quadratic equation in terms of ( c^2 ). Let me set ( u = c^2 ):[ u^2 - 8a^2u + 4a^4 = 0 ]Now, solve for ( u ) using the quadratic formula:[ u = dfrac{8a^2 pm sqrt{(8a^2)^2 - 4 times 1 times 4a^4}}{2} ]Simplify inside the square root:[ (8a^2)^2 = 64a^4 ][ 4 times 1 times 4a^4 = 16a^4 ]So, the discriminant is:[ 64a^4 - 16a^4 = 48a^4 ]Thus:[ u = dfrac{8a^2 pm sqrt{48a^4}}{2} ]Simplify the square root:[ sqrt{48a^4} = sqrt{16 times 3 times a^4} = 4a^2sqrt{3} ]So:[ u = dfrac{8a^2 pm 4a^2sqrt{3}}{2} ]Factor out ( 4a^2 ):[ u = dfrac{4a^2(2 pm sqrt{3})}{2} ]Simplify:[ u = 2a^2(2 pm sqrt{3}) ]So, ( u = 2a^2(2 + sqrt{3}) ) or ( u = 2a^2(2 - sqrt{3}) )But ( u = c^2 ), and since ( c^2 = a^2 + b^2 ), ( c^2 ) must be greater than ( a^2 ). Let's check both solutions:1. ( c^2 = 2a^2(2 + sqrt{3}) ) This is definitely greater than ( a^2 ), so it's valid.2. ( c^2 = 2a^2(2 - sqrt{3}) ) Let's compute ( 2 - sqrt{3} approx 2 - 1.732 = 0.267 ) So, ( c^2 approx 2a^2 times 0.267 approx 0.534a^2 ) But ( c^2 ) must be greater than ( a^2 ), so this is invalid.Therefore, the only valid solution is:[ c^2 = 2a^2(2 + sqrt{3}) ]Simplify:[ c^2 = 4a^2 + 2a^2sqrt{3} ]Now, recall that ( c^2 = a^2 + b^2 ), so:[ a^2 + b^2 = 4a^2 + 2a^2sqrt{3} ]Subtract ( a^2 ) from both sides:[ b^2 = 3a^2 + 2a^2sqrt{3} ]Factor out ( a^2 ):[ b^2 = a^2(3 + 2sqrt{3}) ]Now, the eccentricity ( e ) is ( e = dfrac{c}{a} ). Since ( c^2 = 4a^2 + 2a^2sqrt{3} ), let's compute ( c ):[ c = sqrt{4a^2 + 2a^2sqrt{3}} ]Factor out ( a^2 ):[ c = asqrt{4 + 2sqrt{3}} ]Let me simplify ( sqrt{4 + 2sqrt{3}} ). I recall that expressions like ( sqrt{a + 2sqrt{b}} ) can sometimes be written as ( sqrt{c} + sqrt{d} ) if ( c + d = a ) and ( cd = b ).Let me assume:[ sqrt{4 + 2sqrt{3}} = sqrt{c} + sqrt{d} ]Then:[ (sqrt{c} + sqrt{d})^2 = c + d + 2sqrt{cd} = 4 + 2sqrt{3} ]So, we have:1. ( c + d = 4 )2. ( 2sqrt{cd} = 2sqrt{3} ) => ( sqrt{cd} = sqrt{3} ) => ( cd = 3 )So, we need two numbers ( c ) and ( d ) such that:- ( c + d = 4 )- ( c times d = 3 )This is a system of equations. Let me solve for ( c ) and ( d ).From the first equation, ( d = 4 - c ). Substitute into the second equation:[ c(4 - c) = 3 ][ 4c - c^2 = 3 ][ c^2 - 4c + 3 = 0 ]Factor:[ (c - 1)(c - 3) = 0 ]So, ( c = 1 ) or ( c = 3 ). Therefore, ( d = 3 ) or ( d = 1 ).Thus:[ sqrt{4 + 2sqrt{3}} = sqrt{1} + sqrt{3} = 1 + sqrt{3} ]Therefore, ( c = a(1 + sqrt{3}) ). So, the eccentricity ( e = dfrac{c}{a} = 1 + sqrt{3} ).Looking back at the options, option C is ( sqrt{3} + 1 ), which is the same as ( 1 + sqrt{3} ). So, the answer should be C.Wait, let me double-check my steps to make sure I didn't make a mistake.1. I found that ( x = dfrac{c}{2} ) for point ( P ).2. Substituted into the hyperbola equation and got an equation involving ( c ) and ( a ).3. Solved the quadratic equation and found ( c^2 = 2a^2(2 + sqrt{3}) ).4. Then, simplified ( c = a(1 + sqrt{3}) ), leading to ( e = 1 + sqrt{3} ).Everything seems to check out. I don't see any errors in the calculations. So, I'm confident that the eccentricity is ( 1 + sqrt{3} ), which is option C.**Final Answer**boxed{C}

💡First, I need to determine the total height of the stack each person is standing on by multiplying the height of one plank by the number of planks in their respective stacks.For Eunji, who is standing on 8 A planks:Total height = 2.4 cm * 8 = 19.2 cmFor Namjoon, who is standing on 4 B planks:Total height = 3.2 cm * 4 = 12.8 cmFor Hoseok, who is standing on 5 C planks:Total height = 2.8 cm * 5 = 14 cmAfter calculating the total heights, I compare them to identify who has the lowest height.Eunji: 19.2 cmNamjoon: 12.8 cmHoseok: 14 cmNamjoon has the lowest total height at 12.8 cm.

💡Okay, so I have this inequality to solve: 3x² + x - 2 < 0. Hmm, I remember that quadratic inequalities can be a bit tricky, but let me try to work through it step by step. First, I think I need to find the roots of the quadratic equation 3x² + x - 2 = 0 because the roots will help me determine the intervals where the quadratic expression is positive or negative.Alright, to find the roots, I can use the quadratic formula, which is x = [-b ± √(b² - 4ac)] / (2a). In this equation, a is 3, b is 1, and c is -2. Plugging these values into the formula, I get:x = [-1 ± √(1² - 4*3*(-2))] / (2*3)x = [-1 ± √(1 + 24)] / 6x = [-1 ± √25] / 6x = [-1 ± 5] / 6So, that gives me two solutions:x = (-1 + 5)/6 = 4/6 = 2/3x = (-1 - 5)/6 = -6/6 = -1Okay, so the roots are x = -1 and x = 2/3. These are the points where the quadratic expression equals zero. Now, to determine where the expression is less than zero (negative), I need to analyze the intervals defined by these roots.The number line is divided into three intervals by the roots -1 and 2/3:1. x < -12. -1 < x < 2/33. x > 2/3I need to test each interval to see whether the quadratic expression is positive or negative in that interval.Let's start with the first interval: x < -1. I'll pick a test point, say x = -2.Plugging x = -2 into 3x² + x - 2:3*(-2)² + (-2) - 2 = 3*4 - 2 - 2 = 12 - 4 = 88 is positive, so the expression is positive when x < -1.Next, the interval -1 < x < 2/3. Let's choose x = 0 as a test point.Plugging x = 0 into the expression:3*(0)² + 0 - 2 = 0 + 0 - 2 = -2-2 is negative, so the expression is negative in this interval.Finally, the interval x > 2/3. Let's pick x = 1 as a test point.Plugging x = 1 into the expression:3*(1)² + 1 - 2 = 3 + 1 - 2 = 22 is positive, so the expression is positive when x > 2/3.Since we're looking for where the expression is less than zero, the solution is the interval where the expression is negative, which is -1 < x < 2/3.Wait, but I think I should also consider the behavior of the quadratic. The coefficient of x² is positive (3), so the parabola opens upwards. That means the expression will be negative between the two roots, which aligns with what I found earlier.So, putting it all together, the solution to the inequality 3x² + x - 2 < 0 is all real numbers x such that x is between -1 and 2/3.I think that's it. Let me just double-check my calculations to make sure I didn't make any mistakes. The roots were calculated correctly using the quadratic formula, and the test points in each interval confirmed the sign of the expression. Yeah, I'm confident that the solution is -1 < x < 2/3.

💡First, I need to understand the given ratio of red, blue, and green balls, which is 2:4:6. To simplify this ratio, I can divide each part by 2, resulting in a simplified ratio of 1:2:3.Next, I recognize that the total number of parts in the ratio is 1 + 2 + 3 = 6 parts. The green balls correspond to 3 parts of this ratio, and it's given that there are 36 green balls.To find the value of one part, I divide the number of green balls by the number of parts they represent: 36 ÷ 3 = 12. This means each part of the ratio represents 12 balls.Finally, to find the total number of balls, I multiply the value of one part by the total number of parts: 12 × 6 = 72. Therefore, there are 72 balls in total.

💡Alright, so I have this problem where I need to define a function ( f(x) ) on the interval ( [0, 1] ) with four specific properties, and then determine the value of ( fleft( frac{2}{9} right) ). Let me try to break this down step by step.First, let me list out the properties again to make sure I understand them:1. ( f(0) = 0 ).2. If ( 0 leq x < y leq 1 ), then ( f(x) leq f(y) ). So, the function is non-decreasing.3. ( f(1 - x) = frac{3}{4} - frac{f(x)}{2} ). This seems like a symmetry property.4. ( fleft( frac{x}{3} right) = frac{f(x)}{3} ) for ( 0 leq x leq 1 ). This looks like a scaling property.My goal is to find ( fleft( frac{2}{9} right) ). Hmm, okay. Let's see how I can use these properties to find that value.Starting with property (i), I know that ( f(0) = 0 ). That's straightforward.Property (ii) tells me that the function is non-decreasing. So, if I have two points ( x ) and ( y ) where ( x < y ), then ( f(x) leq f(y) ). This might be useful later when I need to ensure that the values I find make sense in the context of the function's behavior.Property (iii) is interesting. It relates the value of the function at ( 1 - x ) to the value at ( x ). Specifically, ( f(1 - x) = frac{3}{4} - frac{f(x)}{2} ). This seems like a kind of reflection property. Maybe I can use this to find values of the function at points related by this symmetry.Property (iv) is a scaling property. It tells me that if I scale the input ( x ) by a factor of ( frac{1}{3} ), then the output scales by ( frac{1}{3} ) as well. So, ( fleft( frac{x}{3} right) = frac{f(x)}{3} ). This suggests that the function has some self-similar behavior when scaled by powers of 3.Okay, so how can I use these properties to find ( fleft( frac{2}{9} right) )?Let me start by trying to find some key values of the function, maybe at points that are fractions with denominators that are powers of 3, since property (iv) involves scaling by 3. That might help me build up to ( frac{2}{9} ).First, let's find ( f(1) ). I don't have that directly, but maybe I can use property (iii) with ( x = 0 ).If I set ( x = 0 ) in property (iii), I get:( f(1 - 0) = frac{3}{4} - frac{f(0)}{2} )Simplifying:( f(1) = frac{3}{4} - frac{0}{2} = frac{3}{4} )So, ( f(1) = frac{3}{4} ). Good, that's a known value.Next, let's try to find ( fleft( frac{1}{3} right) ). Using property (iv), if I set ( x = 1 ), then:( fleft( frac{1}{3} right) = frac{f(1)}{3} = frac{frac{3}{4}}{3} = frac{1}{4} )So, ( fleft( frac{1}{3} right) = frac{1}{4} ). That's another known value.Now, let's see if I can find ( fleft( frac{2}{3} right) ). Maybe I can use property (iii) again. Let me set ( x = frac{1}{3} ):( fleft( 1 - frac{1}{3} right) = frac{3}{4} - frac{fleft( frac{1}{3} right)}{2} )Simplifying:( fleft( frac{2}{3} right) = frac{3}{4} - frac{frac{1}{4}}{2} = frac{3}{4} - frac{1}{8} = frac{6}{8} - frac{1}{8} = frac{5}{8} )So, ( fleft( frac{2}{3} right) = frac{5}{8} ). That's another value.Now, I need to find ( fleft( frac{2}{9} right) ). Let's see how I can get there. Since ( frac{2}{9} = frac{2}{3} times frac{1}{3} ), maybe I can use property (iv) again.If I set ( x = frac{2}{3} ) in property (iv):( fleft( frac{frac{2}{3}}{3} right) = frac{fleft( frac{2}{3} right)}{3} )Simplifying:( fleft( frac{2}{9} right) = frac{frac{5}{8}}{3} = frac{5}{24} )So, ( fleft( frac{2}{9} right) = frac{5}{24} ).Wait a minute, let me double-check that. I used property (iv) on ( x = frac{2}{3} ) to get ( fleft( frac{2}{9} right) ). That seems correct because ( frac{2}{9} = frac{2}{3} times frac{1}{3} ), so scaling ( frac{2}{3} ) by ( frac{1}{3} ) gives ( frac{2}{9} ), and the function value scales by ( frac{1}{3} ) as well.Also, checking the non-decreasing property (ii), since ( frac{1}{9} < frac{2}{9} < frac{1}{3} ), and we have ( fleft( frac{1}{9} right) = frac{1}{12} ), ( fleft( frac{2}{9} right) = frac{5}{24} ), and ( fleft( frac{1}{3} right) = frac{1}{4} ). Let's convert these to a common denominator to compare:( frac{1}{12} = frac{2}{24} ), ( frac{5}{24} ) stays the same, and ( frac{1}{4} = frac{6}{24} ).So, ( frac{2}{24} < frac{5}{24} < frac{6}{24} ), which means ( fleft( frac{1}{9} right) < fleft( frac{2}{9} right) < fleft( frac{1}{3} right) ). That satisfies the non-decreasing property.Just to be thorough, let me see if there's another way to find ( fleft( frac{2}{9} right) ) using property (iii). Maybe by expressing ( frac{2}{9} ) as ( 1 - x ) for some ( x ), and then using the symmetry property.Let me solve for ( x ) in ( 1 - x = frac{2}{9} ):( x = 1 - frac{2}{9} = frac{7}{9} )So, ( fleft( frac{2}{9} right) = frac{3}{4} - frac{fleft( frac{7}{9} right)}{2} )Hmm, but I don't know ( fleft( frac{7}{9} right) ) yet. Maybe I can find it using property (iv) or (iii). Let's try.First, let's find ( fleft( frac{7}{9} right) ). Since ( frac{7}{9} = frac{7}{3} times frac{1}{3} ), but ( frac{7}{3} ) is greater than 1, which is outside our domain. So, that approach might not work directly.Alternatively, maybe I can express ( frac{7}{9} ) in terms of other fractions. Let's see:( frac{7}{9} = frac{7}{9} ), but that doesn't immediately suggest a way to use property (iv). Maybe I can use property (iii) again on ( frac{7}{9} ).Let me set ( x = frac{7}{9} ) in property (iii):( fleft( 1 - frac{7}{9} right) = frac{3}{4} - frac{fleft( frac{7}{9} right)}{2} )Simplifying:( fleft( frac{2}{9} right) = frac{3}{4} - frac{fleft( frac{7}{9} right)}{2} )But this is the same equation I had earlier, so it's not giving me new information. I need another way to find ( fleft( frac{7}{9} right) ).Wait, maybe I can express ( frac{7}{9} ) as ( frac{7}{9} = frac{7}{3} times frac{1}{3} ), but since ( frac{7}{3} > 1 ), I can't directly apply property (iv). However, perhaps I can relate ( frac{7}{9} ) to another point within the domain.Alternatively, maybe I can find ( fleft( frac{7}{9} right) ) by using property (iv) in reverse. Let me think.If I have ( fleft( frac{x}{3} right) = frac{f(x)}{3} ), then rearranging gives ( f(x) = 3 fleft( frac{x}{3} right) ). So, if I can express ( frac{7}{9} ) as ( frac{x}{3} ), then ( x = frac{7}{3} ), which is outside the domain. So, that doesn't help.Hmm, maybe I need to find another way. Let me consider that ( frac{7}{9} ) is greater than ( frac{2}{3} ), which I already know ( fleft( frac{2}{3} right) = frac{5}{8} ). Since the function is non-decreasing, ( fleft( frac{7}{9} right) geq fleft( frac{2}{3} right) = frac{5}{8} ).But without more information, I can't determine the exact value of ( fleft( frac{7}{9} right) ) directly. So, maybe the first approach I took, using property (iv) on ( frac{2}{3} ), is the correct way to find ( fleft( frac{2}{9} right) ).Let me recap:1. ( f(1) = frac{3}{4} ) from property (iii) with ( x = 0 ).2. ( fleft( frac{1}{3} right) = frac{1}{4} ) from property (iv) with ( x = 1 ).3. ( fleft( frac{2}{3} right) = frac{5}{8} ) from property (iii) with ( x = frac{1}{3} ).4. ( fleft( frac{2}{9} right) = frac{5}{24} ) from property (iv) with ( x = frac{2}{3} ).This seems consistent and I don't see any contradictions with the given properties. Also, checking the non-decreasing property, as I did earlier, confirms that the values are in the correct order.Therefore, I think the value of ( fleft( frac{2}{9} right) ) is ( frac{5}{24} ).