💡First, I need to identify the unknown number and represent it with a variable, let's say ( x ).The problem states that when this number is increased by 2 and then by 8, the result is equal to three times the number. This can be translated into the equation:[x + 2 + 8 = 3x]Next, I'll simplify the left side of the equation by combining the constants:[x + 10 = 3x]To isolate ( x ), I'll subtract ( x ) from both sides of the equation:[10 = 2x]Finally, I'll solve for ( x ) by dividing both sides by 2:[x = 5]

💡First, I need to visualize the regular hexagon and determine how far Sam travels along its perimeter. Each side of the hexagon is 4 km, and Sam walks a total of 10 km. This means he covers two full sides (8 km) and half of the third side (2 km).Next, I'll assign coordinates to the vertices of the hexagon to better understand Sam's path. Starting from the origin (0,0), the first vertex is at (4,0). After walking another 4 km, Sam reaches the second vertex at (6, 2√3). Finally, walking the remaining 2 km along the third side brings him to the point (5, √3).To find the shortest distance from Sam's final position back to the starting point, I'll use the distance formula. The coordinates of the starting point are (0,0), and Sam's final position is (5, √3). Plugging these into the distance formula gives:Distance = √[(5 - 0)² + (√3 - 0)²] = √[25 + 3] = √28 = 2√7 km.Therefore, the shortest distance from where Sam ends up to his starting point is 2√7 kilometers.

💡To determine the total capacity of the water tank, let's analyze the given information step by step.First, when the tank is 40% full, it contains 36 gallons less than when it is 30% empty. Interpret '30% empty' as '70% full', since 100% - 30% = 70%.Let ( T ) be the total capacity of the tank in gallons. Then, ( 0.7T ) and ( 0.4T ) represent the volumes when the tank is 70% full and 40% full, respectively.The difference between 70% full and 40% full is given to be 36 gallons:[0.7T - 0.4T = 0.3T = 36 text{ gallons}]Solve for ( T ):[0.3T = 36][T = frac{36}{0.3} = 120 text{ gallons}]Hence, the total capacity of the water tank is (boxed{120 text{ gallons}}).

💡Alright, so I have this problem here: In triangle XYZ, angle X is 90 degrees, and sin Y is 3/5. I need to find cos Y. Hmm, okay, let's see. I remember that in a right-angled triangle, the sine and cosine of an angle are related to the sides of the triangle. First, let me visualize the triangle. Since angle X is 90 degrees, that means sides XY and XZ are the legs, and YZ is the hypotenuse. So, angle Y is at vertex Y, and angle Z is at vertex Z. Now, sin Y is given as 3/5. I recall that sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. So, for angle Y, the opposite side would be XZ, and the hypotenuse is YZ. That means XZ/YZ = 3/5. Let me denote the lengths of the sides. Let's say the length of XZ is 3k, and the hypotenuse YZ is 5k, where k is some positive real number. This way, the ratio XZ/YZ remains 3/5, which matches the given sine value. Now, I need to find the length of the remaining side, which is XY. Since it's a right-angled triangle, I can use the Pythagorean theorem. The theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, XY² + XZ² = YZ².Substituting the values I have:XY² + (3k)² = (5k)²XY² + 9k² = 25k²XY² = 25k² - 9k²XY² = 16k²Taking the square root of both sides:XY = 4kOkay, so the length of XY is 4k. Now, I need to find cos Y. Cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. For angle Y, the adjacent side is XY, and the hypotenuse is YZ. So, cos Y = XY/YZ.Substituting the values I have:cos Y = (4k)/(5k)The k cancels out, so cos Y = 4/5.Wait, let me double-check that. If sin Y is 3/5, then the sides opposite and hypotenuse are in the ratio 3:5. Using Pythagoras, the other side comes out to 4, so the sides are 3, 4, 5. That makes sense because 3-4-5 is a classic Pythagorean triple. Therefore, cos Y, being adjacent over hypotenuse, should indeed be 4/5.I think that makes sense. So, cos Y is 4/5.

💡Alright, so I have this problem about a rectangle (ABCD) where (AB = 20) and (AD = 24). There's a point (P) inside the rectangle such that the areas of triangles (PAC) and (PBD) are 20 and 24, respectively. I need to find all possible areas of triangle (PAB).First, I should probably draw a diagram to visualize the problem. Let me imagine rectangle (ABCD) with (AB) as the base of length 20 and (AD) as the height of 24. So, (ABCD) has vertices (A), (B), (C), and (D) in order, making it a rectangle.Point (P) is somewhere inside this rectangle. The areas of triangles (PAC) and (PBD) are given as 20 and 24. I need to figure out how these areas relate to the position of (P) and then use that to find the area of triangle (PAB).Let me recall that the area of a triangle is (frac{1}{2} times text{base} times text{height}). Maybe I can use coordinate geometry to solve this. Let me assign coordinates to the rectangle.Let's place point (A) at the origin ((0, 0)). Then, since (AB = 20), point (B) would be at ((20, 0)). Point (D) would be at ((0, 24)), and point (C), being the opposite corner from (A), would be at ((20, 24)).So, coordinates:- (A = (0, 0))- (B = (20, 0))- (C = (20, 24))- (D = (0, 24))Now, point (P) is inside the rectangle, so its coordinates are ((x, y)) where (0 < x < 20) and (0 < y < 24).Now, let's find the areas of triangles (PAC) and (PBD) in terms of (x) and (y).Starting with triangle (PAC). The vertices of triangle (PAC) are (P(x, y)), (A(0, 0)), and (C(20, 24)). The area can be calculated using the shoelace formula or the determinant method.Using the determinant formula for the area of a triangle with vertices ((x_1, y_1)), ((x_2, y_2)), ((x_3, y_3)):[text{Area} = frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|]Plugging in the coordinates for (PAC):[text{Area}_{PAC} = frac{1}{2} |x(0 - 24) + 0(24 - y) + 20(y - 0)|][= frac{1}{2} |-24x + 0 + 20y|][= frac{1}{2} | -24x + 20y |]Given that the area is 20, we have:[frac{1}{2} | -24x + 20y | = 20]Multiply both sides by 2:[| -24x + 20y | = 40]So,[-24x + 20y = 40 quad text{or} quad -24x + 20y = -40]Simplify both equations:1. (-24x + 20y = 40) Divide both sides by 4: (-6x + 5y = 10)2. (-24x + 20y = -40) Divide both sides by 4: (-6x + 5y = -10)So, we have two equations:1. (-6x + 5y = 10)2. (-6x + 5y = -10)Now, let's do the same for triangle (PBD). The vertices of triangle (PBD) are (P(x, y)), (B(20, 0)), and (D(0, 24)).Using the determinant formula again:[text{Area}_{PBD} = frac{1}{2} |x(0 - 24) + 20(24 - y) + 0(y - 0)|][= frac{1}{2} |-24x + 20(24 - y) + 0|][= frac{1}{2} |-24x + 480 - 20y|]Given that the area is 24:[frac{1}{2} | -24x - 20y + 480 | = 24]Multiply both sides by 2:[| -24x - 20y + 480 | = 48]So,[-24x - 20y + 480 = 48 quad text{or} quad -24x - 20y + 480 = -48]Simplify both equations:1. (-24x - 20y + 480 = 48) Subtract 480 from both sides: (-24x - 20y = -432) Divide both sides by -4: (6x + 5y = 108)2. (-24x - 20y + 480 = -48) Subtract 480 from both sides: (-24x - 20y = -528) Divide both sides by -4: (6x + 5y = 132)So, we have two more equations:3. (6x + 5y = 108)4. (6x + 5y = 132)Now, we have four possible combinations of equations from the two areas:1. (-6x + 5y = 10) and (6x + 5y = 108)2. (-6x + 5y = 10) and (6x + 5y = 132)3. (-6x + 5y = -10) and (6x + 5y = 108)4. (-6x + 5y = -10) and (6x + 5y = 132)Let me solve each pair to find the coordinates of (P).**Case 1:**(-6x + 5y = 10) and (6x + 5y = 108)Add both equations:((-6x + 5y) + (6x + 5y) = 10 + 108)Simplify:(10y = 118)So,(y = 11.8)Now, substitute (y = 11.8) into one of the equations, say (-6x + 5y = 10):(-6x + 5(11.8) = 10)Calculate:(-6x + 59 = 10)Subtract 59:(-6x = -49)Divide by -6:(x = frac{49}{6} approx 8.1667)So, (P) is at ((frac{49}{6}, 11.8)).**Case 2:**(-6x + 5y = 10) and (6x + 5y = 132)Add both equations:(10y = 142)So,(y = 14.2)Substitute (y = 14.2) into (-6x + 5y = 10):(-6x + 5(14.2) = 10)Calculate:(-6x + 71 = 10)Subtract 71:(-6x = -61)Divide by -6:(x = frac{61}{6} approx 10.1667)So, (P) is at ((frac{61}{6}, 14.2)).**Case 3:**(-6x + 5y = -10) and (6x + 5y = 108)Add both equations:(10y = 98)So,(y = 9.8)Substitute (y = 9.8) into (-6x + 5y = -10):(-6x + 5(9.8) = -10)Calculate:(-6x + 49 = -10)Subtract 49:(-6x = -59)Divide by -6:(x = frac{59}{6} approx 9.8333)So, (P) is at ((frac{59}{6}, 9.8)).**Case 4:**(-6x + 5y = -10) and (6x + 5y = 132)Add both equations:(10y = 122)So,(y = 12.2)Substitute (y = 12.2) into (-6x + 5y = -10):(-6x + 5(12.2) = -10)Calculate:(-6x + 61 = -10)Subtract 61:(-6x = -71)Divide by -6:(x = frac{71}{6} approx 11.8333)So, (P) is at ((frac{71}{6}, 12.2)).Now, I have four possible coordinates for point (P):1. ((frac{49}{6}, 11.8))2. ((frac{61}{6}, 14.2))3. ((frac{59}{6}, 9.8))4. ((frac{71}{6}, 12.2))Next, I need to compute the area of triangle (PAB) for each of these points.Triangle (PAB) has vertices (P(x, y)), (A(0, 0)), and (B(20, 0)).Using the determinant formula again:[text{Area}_{PAB} = frac{1}{2} |x(0 - 0) + 0(0 - y) + 20(y - 0)|]Simplify:[= frac{1}{2} |0 + 0 + 20y|][= frac{1}{2} |20y|][= 10y]So, the area of triangle (PAB) is simply (10y), where (y) is the y-coordinate of point (P).Therefore, for each point (P), the area is:1. For (P(frac{49}{6}, 11.8)): (10 times 11.8 = 118)2. For (P(frac{61}{6}, 14.2)): (10 times 14.2 = 142)3. For (P(frac{59}{6}, 9.8)): (10 times 9.8 = 98)4. For (P(frac{71}{6}, 12.2)): (10 times 12.2 = 122)So, the possible areas of triangle (PAB) are 98, 118, 122, and 142.Wait a second, let me double-check if all these points are inside the rectangle. Since (0 < x < 20) and (0 < y < 24), all the y-values are between 9.8 and 14.2, which is within the rectangle's height of 24. The x-values are approximately 8.1667, 10.1667, 9.8333, and 11.8333, all within 0 and 20. So, all four points are indeed inside the rectangle.Therefore, all four areas are valid.**Final Answer**The possible areas of triangle (PAB) are boxed{98}, boxed{118}, boxed{122}, and boxed{142}.

💡Okay, so I have this problem here: The symmetry axis of the function ( y = a sin x - b cos x ) is ( x = frac{pi}{4} ). Then, I need to find the inclination angle of the line ( l: ax - by + c = 0 ). The options are 45°, 60°, 120°, and 135°. Hmm, let me try to figure this out step by step.First, I remember that a function having a symmetry axis at ( x = frac{pi}{4} ) means that the function is symmetric around that line. So, for any ( x ), ( fleft( frac{pi}{4} + x right) = fleft( frac{pi}{4} - x right) ). That makes sense. So, I can set up the equation:( a sinleft( frac{pi}{4} + x right) - b cosleft( frac{pi}{4} + x right) = a sinleft( frac{pi}{4} - x right) - b cosleft( frac{pi}{4} - x right) )Now, I need to simplify both sides. Let me recall the sine and cosine addition formulas:( sin(A pm B) = sin A cos B pm cos A sin B )( cos(A pm B) = cos A cos B mp sin A sin B )So, applying these to both sides:Left side:( a left[ sin frac{pi}{4} cos x + cos frac{pi}{4} sin x right] - b left[ cos frac{pi}{4} cos x - sin frac{pi}{4} sin x right] )Right side:( a left[ sin frac{pi}{4} cos x - cos frac{pi}{4} sin x right] - b left[ cos frac{pi}{4} cos x + sin frac{pi}{4} sin x right] )Let me compute both sides step by step.First, compute the left side:( a sin frac{pi}{4} cos x + a cos frac{pi}{4} sin x - b cos frac{pi}{4} cos x + b sin frac{pi}{4} sin x )Similarly, compute the right side:( a sin frac{pi}{4} cos x - a cos frac{pi}{4} sin x - b cos frac{pi}{4} cos x - b sin frac{pi}{4} sin x )Now, let me subtract the right side from the left side to set up the equation:Left side - Right side:( [a sin frac{pi}{4} cos x + a cos frac{pi}{4} sin x - b cos frac{pi}{4} cos x + b sin frac{pi}{4} sin x] - [a sin frac{pi}{4} cos x - a cos frac{pi}{4} sin x - b cos frac{pi}{4} cos x - b sin frac{pi}{4} sin x] = 0 )Simplify term by term:1. ( a sin frac{pi}{4} cos x - a sin frac{pi}{4} cos x = 0 )2. ( a cos frac{pi}{4} sin x - (-a cos frac{pi}{4} sin x) = 2a cos frac{pi}{4} sin x )3. ( -b cos frac{pi}{4} cos x - (-b cos frac{pi}{4} cos x) = 0 )4. ( b sin frac{pi}{4} sin x - (-b sin frac{pi}{4} sin x) = 2b sin frac{pi}{4} sin x )So, combining these:( 2a cos frac{pi}{4} sin x + 2b sin frac{pi}{4} sin x = 0 )Factor out the common terms:( 2 sin x left( a cos frac{pi}{4} + b sin frac{pi}{4} right) = 0 )Since this must hold for any ( x ), the coefficient of ( sin x ) must be zero:( a cos frac{pi}{4} + b sin frac{pi}{4} = 0 )I know that ( cos frac{pi}{4} = sin frac{pi}{4} = frac{sqrt{2}}{2} ), so substituting:( a cdot frac{sqrt{2}}{2} + b cdot frac{sqrt{2}}{2} = 0 )Factor out ( frac{sqrt{2}}{2} ):( frac{sqrt{2}}{2} (a + b) = 0 )Since ( frac{sqrt{2}}{2} ) is not zero, we have:( a + b = 0 )So, ( a = -b ). That's a key relationship between ( a ) and ( b ).Now, moving on to the line ( l: ax - by + c = 0 ). I need to find its inclination angle. The inclination angle is the angle that the line makes with the positive direction of the x-axis, measured counterclockwise. It's given by ( theta ) where ( tan theta = m ), the slope of the line.First, let me write the equation of the line in slope-intercept form ( y = mx + k ). Starting with ( ax - by + c = 0 ):( ax + c = by )( y = frac{a}{b} x + frac{c}{b} )So, the slope ( m = frac{a}{b} ).But from earlier, we found that ( a = -b ). So, substituting:( m = frac{-b}{b} = -1 )So, the slope is ( -1 ). Now, the inclination angle ( theta ) satisfies ( tan theta = -1 ). Hmm, but the inclination angle is defined as the smallest non-negative angle between the positive x-axis and the line, measured counterclockwise. So, it should be between 0° and 180°, I think.Wait, actually, the inclination angle is usually defined as the angle between 0° and 180°, right? So, if the slope is negative, the angle is measured from the positive x-axis going upwards to the line, which would be in the second quadrant.So, ( tan theta = -1 ). The reference angle for this is 45°, but since the slope is negative, the angle is 180° - 45° = 135°. So, the inclination angle is 135°, which is option D.Wait, let me double-check. If the slope is negative, the line is going downwards from left to right. So, the angle with the positive x-axis is indeed 135°, which is in the second quadrant. That makes sense.Just to recap:1. The function ( y = a sin x - b cos x ) has a symmetry axis at ( x = frac{pi}{4} ).2. This implies that ( fleft( frac{pi}{4} + x right) = fleft( frac{pi}{4} - x right) ) for all ( x ).3. Expanding both sides using trigonometric identities and simplifying leads to the condition ( a + b = 0 ).4. For the line ( ax - by + c = 0 ), the slope is ( frac{a}{b} = -1 ).5. The inclination angle corresponding to a slope of -1 is 135°.Therefore, the correct answer should be D: 135°.**Final Answer**boxed{D}