💡First, I need to determine the time taken for each half of the journey. The total journey is 560 km, so each half is 280 km.For the first half, the woman travels at 21 km/hr. Using the formula Time = Distance / Speed, the time for the first half is 280 km divided by 21 km/hr, which equals approximately 13.33 hours.For the second half, she travels at 24 km/hr. Using the same formula, the time for the second half is 280 km divided by 24 km/hr, which equals approximately 11.67 hours.Finally, I add the times for both halves to find the total journey time: 13.33 hours plus 11.67 hours equals 25 hours.

💡To solve the equation ( a + bi = i^{2019} ), we start by simplifying ( i^{2019} ).First, we recognize that the powers of ( i ) cycle every 4:[i^1 = i,quad i^2 = -1,quad i^3 = -i,quad i^4 = 1]Thus, we can express ( i^{2019} ) as:[i^{2019} = (i^4)^{504} cdot i^3 = 1^{504} cdot (-i) = -i]This means:[a + bi = -i]By comparing the real and imaginary parts, we find:[a = 0 quad text{and} quad b = -1]Finally, we calculate ( a + b ):[a + b = 0 + (-1) = -1]

💡Okay, so I have this problem about a parabola and a line intersecting at points A and B. The parabola is given by the equation ( y^2 = 4x ) and the line is ( y = 2x - 4 ). There are two parts: first, I need to find the length of the chord AB, and second, I need to find the coordinates of a point P on the parabola such that the area of triangle ABP is 12.Starting with part (1). I remember that to find the points of intersection between a line and a parabola, I can substitute the equation of the line into the equation of the parabola. So, since ( y = 2x - 4 ), I can substitute this into ( y^2 = 4x ).Let me write that out:( (2x - 4)^2 = 4x )Expanding the left side:( 4x^2 - 16x + 16 = 4x )Now, subtract 4x from both sides to set the equation to zero:( 4x^2 - 16x + 16 - 4x = 0 )Simplify the terms:( 4x^2 - 20x + 16 = 0 )Hmm, this is a quadratic equation in terms of x. Maybe I can simplify it by dividing all terms by 4 to make it easier:( x^2 - 5x + 4 = 0 )Okay, now I can factor this quadratic:Looking for two numbers that multiply to 4 and add up to -5. That would be -1 and -4.So, ( (x - 1)(x - 4) = 0 )Therefore, the solutions are x = 1 and x = 4.Now, plugging these back into the equation of the line ( y = 2x - 4 ) to find the corresponding y-values.For x = 1:( y = 2(1) - 4 = 2 - 4 = -2 )So, point A is (1, -2).For x = 4:( y = 2(4) - 4 = 8 - 4 = 4 )So, point B is (4, 4).Now, I need to find the length of chord AB. The distance between points A(1, -2) and B(4, 4).Using the distance formula:( AB = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} )Plugging in the values:( AB = sqrt{(4 - 1)^2 + (4 - (-2))^2} )Simplify:( AB = sqrt{(3)^2 + (6)^2} = sqrt{9 + 36} = sqrt{45} )Simplify the square root:( sqrt{45} = sqrt{9 times 5} = 3sqrt{5} )So, the length of chord AB is ( 3sqrt{5} ).Wait, that seems straightforward. Let me double-check my steps.1. Substituted the line equation into the parabola equation correctly.2. Expanded and simplified the quadratic correctly.3. Factored the quadratic correctly to find x = 1 and x = 4.4. Plugged back into the line equation to find y-values, which gave me points (1, -2) and (4, 4).5. Applied the distance formula correctly between these two points.Everything seems to check out. So, part (1) is done.Moving on to part (2). I need to find the coordinates of point P on the parabola ( y^2 = 4x ) such that the area of triangle ABP is 12.First, I recall that the area of a triangle given three points can be found using the shoelace formula or the determinant method. Alternatively, since I know the coordinates of A and B, and I know the area, maybe I can use the formula involving the base and height.But since point P is variable on the parabola, perhaps it's better to use the formula for the area of a triangle with vertices at (x1, y1), (x2, y2), and (x3, y3):( text{Area} = frac{1}{2} | (x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)) | )Alternatively, another formula is:( text{Area} = frac{1}{2} | vec{AB} times vec{AP} | )But maybe using the determinant method is more straightforward here.Given points A(1, -2), B(4, 4), and P(x, y) on the parabola ( y^2 = 4x ), so x = ( frac{y^2}{4} ).So, point P can be represented as ( left( frac{y^2}{4}, y right) ).Let me denote P as ( left( frac{y^2}{4}, y right) ).Now, let's set up the area formula using the determinant method.The area of triangle ABP is:( text{Area} = frac{1}{2} | x_A(y_B - y_P) + x_B(y_P - y_A) + x_P(y_A - y_B) | )Plugging in the coordinates:( text{Area} = frac{1}{2} | 1(4 - y) + 4(y - (-2)) + frac{y^2}{4}(-2 - 4) | )Simplify each term:First term: ( 1(4 - y) = 4 - y )Second term: ( 4(y + 2) = 4y + 8 )Third term: ( frac{y^2}{4}(-6) = -frac{6y^2}{4} = -frac{3y^2}{2} )Combine all terms:( 4 - y + 4y + 8 - frac{3y^2}{2} )Simplify:Combine like terms:- Constants: 4 + 8 = 12- y terms: -y + 4y = 3ySo, we have:( 12 + 3y - frac{3y^2}{2} )Factor out the 3:( 3left(4 + y - frac{y^2}{2}right) )But let me write it as:( 12 + 3y - frac{3y^2}{2} )So, putting it back into the area formula:( text{Area} = frac{1}{2} | 12 + 3y - frac{3y^2}{2} | )We are told that the area is 12, so:( frac{1}{2} | 12 + 3y - frac{3y^2}{2} | = 12 )Multiply both sides by 2:( | 12 + 3y - frac{3y^2}{2} | = 24 )So, the expression inside the absolute value can be either 24 or -24.Therefore, we have two equations:1. ( 12 + 3y - frac{3y^2}{2} = 24 )2. ( 12 + 3y - frac{3y^2}{2} = -24 )Let me solve each equation separately.Starting with equation 1:( 12 + 3y - frac{3y^2}{2} = 24 )Subtract 24 from both sides:( 12 + 3y - frac{3y^2}{2} - 24 = 0 )Simplify:( -12 + 3y - frac{3y^2}{2} = 0 )Multiply all terms by 2 to eliminate the fraction:( -24 + 6y - 3y^2 = 0 )Rearrange terms:( -3y^2 + 6y - 24 = 0 )Multiply both sides by -1:( 3y^2 - 6y + 24 = 0 )Divide all terms by 3:( y^2 - 2y + 8 = 0 )Now, discriminant D = ( (-2)^2 - 4(1)(8) = 4 - 32 = -28 )Since the discriminant is negative, there are no real solutions for this equation. So, equation 1 doesn't yield any real points P.Now, moving on to equation 2:( 12 + 3y - frac{3y^2}{2} = -24 )Add 24 to both sides:( 12 + 3y - frac{3y^2}{2} + 24 = 0 )Simplify:( 36 + 3y - frac{3y^2}{2} = 0 )Multiply all terms by 2:( 72 + 6y - 3y^2 = 0 )Rearrange terms:( -3y^2 + 6y + 72 = 0 )Multiply both sides by -1:( 3y^2 - 6y - 72 = 0 )Divide all terms by 3:( y^2 - 2y - 24 = 0 )Now, solve for y using quadratic formula:( y = frac{2 pm sqrt{(-2)^2 - 4(1)(-24)}}{2(1)} )Simplify inside the square root:( sqrt{4 + 96} = sqrt{100} = 10 )So,( y = frac{2 pm 10}{2} )Calculating both possibilities:1. ( y = frac{2 + 10}{2} = frac{12}{2} = 6 )2. ( y = frac{2 - 10}{2} = frac{-8}{2} = -4 )So, the possible y-values for point P are 6 and -4.Now, find the corresponding x-values on the parabola ( y^2 = 4x ).For y = 6:( x = frac{6^2}{4} = frac{36}{4} = 9 )So, point P is (9, 6).For y = -4:( x = frac{(-4)^2}{4} = frac{16}{4} = 4 )So, point P is (4, -4).Therefore, the coordinates of point P are (9, 6) and (4, -4).Wait, let me verify if these points indeed give the area of 12.First, take point (9, 6):Compute the area using the determinant formula:( text{Area} = frac{1}{2} |1(4 - 6) + 4(6 - (-2)) + 9(-2 - 4)| )Simplify:( frac{1}{2} |1(-2) + 4(8) + 9(-6)| )Calculate each term:- ( 1(-2) = -2 )- ( 4(8) = 32 )- ( 9(-6) = -54 )Add them up:( -2 + 32 - 54 = (-2 - 54) + 32 = -56 + 32 = -24 )Take absolute value and multiply by 1/2:( frac{1}{2} | -24 | = frac{1}{2} times 24 = 12 )Good, that works.Now, for point (4, -4):Compute the area:( text{Area} = frac{1}{2} |1(4 - (-4)) + 4(-4 - (-2)) + 4(-2 - 4)| )Simplify:( frac{1}{2} |1(8) + 4(-2) + 4(-6)| )Calculate each term:- ( 1(8) = 8 )- ( 4(-2) = -8 )- ( 4(-6) = -24 )Add them up:( 8 - 8 - 24 = (8 - 8) - 24 = 0 - 24 = -24 )Take absolute value and multiply by 1/2:( frac{1}{2} | -24 | = frac{1}{2} times 24 = 12 )Perfect, that also works.So, both points (9, 6) and (4, -4) satisfy the area condition.Therefore, the coordinates of point P are (9, 6) and (4, -4).I think that's all. Let me recap:1. Found points A and B by solving the intersection of the parabola and the line.2. Calculated the distance between A and B to find the chord length.3. For part (2), used the area formula for a triangle with coordinates, substituted point P in terms of y, solved the resulting quadratic equations, and found the valid y-values.4. Checked both solutions to ensure they give the correct area.Everything seems consistent. I don't think I made any calculation errors, but let me quickly verify the quadratic solutions again.For equation 1: ended up with discriminant negative, so no real solutions. Correct.For equation 2: quadratic gave y = 6 and y = -4, which led to points (9,6) and (4,-4). Correct.Yes, I think that's solid.**Final Answer**(1) The length of chord ( AB ) is (boxed{3sqrt{5}}).(2) The coordinates of point ( P ) are (boxed{(9, 6)}) and (boxed{(4, -4)}).

💡First, I recognize that in a division sum, the relationship between the dividend, divisor, quotient, and remainder is given by the equation:Dividend = (Divisor × Quotient) + RemainderGiven that the dividend is 1565, the divisor is 24, and the remainder is 5, I can set up the equation as:1565 = (24 × Quotient) + 5To find the quotient, I'll first subtract the remainder from the dividend:1565 - 5 = 1560Now, I'll divide this result by the divisor to determine the quotient:1560 ÷ 24 = 65Therefore, the quotient is 65.

💡Alright, so I have this problem here about vectors. It says that there are two vectors, a and b. The magnitude of vector a is 1, and the magnitude of vector b is 6. There's also this equation: a dot (b minus a) equals 2. I need to find the value of a dot b and the angle between vectors a and b.Okay, let's break this down. First, I remember that the dot product of two vectors can be expressed in terms of their magnitudes and the cosine of the angle between them. The formula is a · b = |a||b|cosθ, where θ is the angle between them. But before I get to that, I need to find a · b.The problem gives me a · (b - a) = 2. Hmm, I think I can expand this expression using the distributive property of the dot product. So, a · (b - a) should be equal to a · b - a · a. That makes sense because the dot product is linear.So, rewriting the equation: a · b - a · a = 2. Now, I know that a · a is just the magnitude of a squared. Since |a| is 1, a · a = 1² = 1. So, substituting that in, the equation becomes a · b - 1 = 2.To solve for a · b, I just need to add 1 to both sides of the equation. That gives me a · b = 3. Okay, so that's the first part done. The dot product of a and b is 3.Now, moving on to finding the angle between them. I remember the formula I mentioned earlier: a · b = |a||b|cosθ. I have a · b, which is 3, and I know |a| is 1 and |b| is 6. Plugging these values into the formula, I get 3 = 1 * 6 * cosθ.Simplifying that, 3 = 6cosθ. To solve for cosθ, I divide both sides by 6, which gives me cosθ = 3/6 = 1/2. So, cosθ equals 1/2.Now, I need to find the angle θ whose cosine is 1/2. From trigonometry, I recall that cos(60°) = 1/2, and since we're dealing with vectors, the angle should be between 0 and 180 degrees. So, θ = 60 degrees, which is π/3 radians.Let me just double-check my steps to make sure I didn't make any mistakes. I expanded the dot product correctly, substituted the known magnitudes, solved for a · b, and then used the dot product formula to find the angle. Everything seems to add up. I think I've got it right.**Final Answer**The value of overrightarrow{a} cdot overrightarrow{b} is boxed{3} and the angle between overrightarrow{a} and overrightarrow{b} is boxed{dfrac{pi}{3}}.

💡First, determine the number of white marbles by subtracting the number of blue and red marbles from the total number of marbles.Next, calculate the total number of red and white marbles.Finally, divide the total number of red and white marbles by the total number of marbles to find the probability.