💡To express (0.000156) in scientific notation, I need to move the decimal point to the right until it is after the first non-zero digit. Starting with (0.000156), I move the decimal point four places to the right, resulting in (1.56). Since I moved the decimal point four places to the right, I multiply by (10^{-4}) to account for the movement. Therefore, (0.000156) in scientific notation is (1.56 times 10^{-4}).

💡First, calculate the total cost of the Spam and the bread.For the Spam:12 cans * 3 per can = 36For the bread:4 loaves * 2 per loaf = 8Add the costs of Spam and bread:36 + 8 = 44Subtract this amount from the total spent to find the cost of peanut butter:59 - 44 = 15Determine the number of peanut butter jars:15 / 5 per jar = 3 jars

💡First, I need to determine Mr. Huang's taxable income by subtracting the tax-free portion from his monthly salary. His salary is 2900 yuan, and the tax-free portion is 2000 yuan. So, the taxable income is 2900 - 2000 = 900 yuan.Next, I'll calculate the personal income tax based on the given tax brackets. The first 500 yuan of the taxable income is taxed at 5%, and the remaining 400 yuan (900 - 500) is taxed at 10%.Calculating the tax for the first portion: 5% of 500 yuan is 25 yuan.Calculating the tax for the second portion: 10% of 400 yuan is 40 yuan.Adding both amounts together gives the total tax: 25 + 40 = 65 yuan.Therefore, Mr. Huang's total personal income tax is 65 yuan, which corresponds to option C.

💡Okay, so I have this problem in the Cartesian coordinate system. It says that there are points A and B on the circle defined by the equation x² + (y - 1)² = 4, and these points A and B are symmetric about the point P(1, 2). I need to find the equation of the line AB.First, I should recall what it means for two points to be symmetric about another point. If points A and B are symmetric about point P, then P is the midpoint of the segment AB. That means the coordinates of P are the average of the coordinates of A and B. So, if A is (x₁, y₁) and B is (x₂, y₂), then ( (x₁ + x₂)/2, (y₁ + y₂)/2 ) should equal (1, 2). So, that gives me two equations: (x₁ + x₂)/2 = 1 and (y₁ + y₂)/2 = 2. Simplifying these, I get x₁ + x₂ = 2 and y₁ + y₂ = 4.Next, I know that both points A and B lie on the circle x² + (y - 1)² = 4. So, plugging in the coordinates of A and B into the circle's equation, I have:For point A: x₁² + (y₁ - 1)² = 4 For point B: x₂² + (y₂ - 1)² = 4Since I know that x₂ = 2 - x₁ and y₂ = 4 - y₁ from the midpoint conditions, I can substitute these into the equation for point B:(2 - x₁)² + (4 - y₁ - 1)² = 4 Simplify that: (2 - x₁)² + (3 - y₁)² = 4Now, I can expand both equations:For point A: x₁² + (y₁ - 1)² = 4 Expanding: x₁² + y₁² - 2y₁ + 1 = 4 Simplify: x₁² + y₁² - 2y₁ = 3For point B: (2 - x₁)² + (3 - y₁)² = 4 Expanding: (4 - 4x₁ + x₁²) + (9 - 6y₁ + y₁²) = 4 Simplify: 4 - 4x₁ + x₁² + 9 - 6y₁ + y₁² = 4 Combine like terms: x₁² + y₁² - 4x₁ - 6y₁ + 13 = 4 Simplify further: x₁² + y₁² - 4x₁ - 6y₁ = -9Now, I have two equations:1. x₁² + y₁² - 2y₁ = 3 2. x₁² + y₁² - 4x₁ - 6y₁ = -9If I subtract the first equation from the second, I can eliminate x₁² and y₁²:(x₁² + y₁² - 4x₁ - 6y₁) - (x₁² + y₁² - 2y₁) = -9 - 3 Simplify: -4x₁ - 4y₁ = -12 Divide both sides by -4: x₁ + y₁ = 3So, this tells me that the coordinates of point A satisfy x + y = 3. Since point A lies on the line AB, and point B is symmetric to A about P, the entire line AB must satisfy this equation. Therefore, the equation of line AB is x + y = 3, or rearranged as x + y - 3 = 0.Wait, let me double-check that. If x + y = 3, then the slope of this line would be -1. Is that consistent with the circle and the point P?The circle is centered at (0, 1) with a radius of 2. The point P is at (1, 2). The line AB is supposed to be such that P is the midpoint. If I draw the line AB with slope -1, it should pass through P(1, 2). Let me verify that.The equation x + y = 3 passes through (1, 2) because 1 + 2 = 3. So that checks out. Also, the line should be perpendicular to the line connecting the center of the circle to the point P. The center is at (0, 1), and P is at (1, 2). The slope of the line connecting (0, 1) to (1, 2) is (2 - 1)/(1 - 0) = 1. So, the slope of AB should be the negative reciprocal, which is -1. That matches our equation.Therefore, I think the equation of line AB is x + y - 3 = 0.**Final Answer**The equation of line AB is boxed{x + y - 3 = 0}.

💡Okay, so I have this problem about a rectangle ABCD with an area of 1 square unit. I need to find the area of a new rectangle that has vertices O and D as opposite corners, where O is the center of the incircle of triangle ABC. The other two vertices of this new rectangle lie on the perimeter of the original rectangle. Hmm, that sounds a bit complicated, but let me try to break it down step by step.First, let me visualize the original rectangle ABCD. Since it's a rectangle, opposite sides are equal and all angles are right angles. The area is 1, so if I let the length be 'a' and the width be 'b', then a * b = 1. That's straightforward.Now, the problem mentions triangle ABC. Since ABCD is a rectangle, triangle ABC is a right-angled triangle with the right angle at B. The vertices of this triangle are A, B, and C. So, triangle ABC has sides AB, BC, and AC. AB is one side of the rectangle, BC is another side, and AC is the diagonal of the rectangle.Next, O is the center of the incircle of triangle ABC. The incircle of a triangle is the largest circle that fits inside the triangle, tangent to all three sides. The center of this incircle is called the incenter, and it's the point where the angle bisectors of the triangle meet. For a right-angled triangle, I remember that the inradius can be calculated using the formula r = (a + b - c)/2, where 'a' and 'b' are the legs and 'c' is the hypotenuse.In this case, the legs of triangle ABC are AB and BC, which are 'a' and 'b' respectively. The hypotenuse AC can be found using the Pythagorean theorem: AC = sqrt(a² + b²). So, the inradius r would be (a + b - sqrt(a² + b²))/2. That gives me the radius of the incircle, but I need the coordinates of the incenter O.Since O is the incenter, it should be located at a point where it is equidistant from all sides of the triangle. In a right-angled triangle, the inradius is located at a distance 'r' from each of the sides. So, if I consider the coordinate system with point B at the origin (0,0), point A at (0, b), point C at (a, 0), and point D at (a, b), then the incenter O can be found.Wait, let me set up a coordinate system to make this clearer. Let's place point B at (0,0). Then, since ABCD is a rectangle, point A would be at (0, b), point C at (a, 0), and point D at (a, b). Triangle ABC has vertices at A(0, b), B(0, 0), and C(a, 0). The incenter O of triangle ABC can be found using the formula for the incenter coordinates, which is ( (a1*x1 + a2*x2 + a3*x3)/perimeter, (a1*y1 + a2*y2 + a3*y3)/perimeter ), where a1, a2, a3 are the lengths of the sides opposite to vertices A, B, C respectively.Wait, no, that's not quite right. The incenter coordinates are actually given by ( (a*x1 + b*x2 + c*x3)/(a + b + c), (a*y1 + b*y2 + c*y3)/(a + b + c) ), where a, b, c are the lengths of the sides opposite to vertices A, B, C respectively.In triangle ABC, side opposite to A is BC, which has length 'a'. Side opposite to B is AC, which is sqrt(a² + b²). Side opposite to C is AB, which has length 'b'. So, the incenter coordinates would be:x = (a*Ax + b*Bx + c*Cx)/(a + b + c)y = (a*Ay + b*By + c*Cy)/(a + b + c)Plugging in the coordinates:Ax = 0, Ay = bBx = 0, By = 0Cx = a, Cy = 0So,x = (a*0 + sqrt(a² + b²)*0 + b*a)/(a + b + sqrt(a² + b²)) = (b*a)/(a + b + sqrt(a² + b²))y = (a*b + sqrt(a² + b²)*0 + b*0)/(a + b + sqrt(a² + b²)) = (a*b)/(a + b + sqrt(a² + b²))So, the coordinates of O are ( (a*b)/(a + b + sqrt(a² + b²)), (a*b)/(a + b + sqrt(a² + b²)) )Hmm, that's interesting. Both x and y coordinates are the same, which makes sense because in a right-angled triangle, the inradius is located at (r, r) where r is the inradius.Wait, let me verify that. Earlier, I had the inradius r = (a + b - c)/2, where c is the hypotenuse. So, r = (a + b - sqrt(a² + b²))/2. So, if I compute (a*b)/(a + b + sqrt(a² + b²)), does that equal r?Let me check:r = (a + b - sqrt(a² + b²))/2Let me compute (a*b)/(a + b + sqrt(a² + b²)):Multiply numerator and denominator by (a + b - sqrt(a² + b²)):(a*b)*(a + b - sqrt(a² + b²)) / [ (a + b + sqrt(a² + b²))(a + b - sqrt(a² + b²)) ]The denominator becomes (a + b)^2 - (sqrt(a² + b²))^2 = a² + 2ab + b² - (a² + b²) = 2abSo, the expression becomes:(a*b)*(a + b - sqrt(a² + b²)) / (2ab) = (a + b - sqrt(a² + b²))/2 = rYes, so (a*b)/(a + b + sqrt(a² + b²)) = r. Therefore, the coordinates of O are (r, r).So, O is at (r, r) where r = (a + b - sqrt(a² + b²))/2.Now, I need to find the area of the new rectangle with vertices O and D as opposite corners, and the other two vertices lying on the perimeter of the original rectangle.Point D is at (a, b). So, the new rectangle has opposite corners at O(r, r) and D(a, b). The other two vertices lie on the perimeter of the original rectangle.Let me visualize this. The original rectangle has sides from (0,0) to (a,0) to (a,b) to (0,b) to (0,0). The new rectangle has one corner at O(r, r) and the opposite corner at D(a, b). The other two vertices must lie on the sides of the original rectangle.Since it's a rectangle, the sides must be parallel to the axes or not? Wait, no, the sides can be at any angle, but since it's a rectangle, the sides must be perpendicular.But actually, in this case, since O and D are opposite corners, the sides of the new rectangle would be parallel to the lines connecting O and D. Wait, no, that's not necessarily true. A rectangle can be rotated, but in this case, since the other two vertices lie on the perimeter of the original rectangle, which is axis-aligned, the new rectangle might have sides that are not parallel to the axes.Hmm, this is getting a bit complicated. Maybe I should parameterize the problem.Let me denote the coordinates:O = (r, r)D = (a, b)Let me denote the other two vertices of the new rectangle as P and Q, lying on the perimeter of the original rectangle.Since it's a rectangle, the vectors OP and OQ should be perpendicular and have the same magnitude as the sides of the rectangle.Wait, maybe it's better to think in terms of coordinates.Let me denote the new rectangle as OPQD, where P and Q lie on the perimeter of ABCD.Since it's a rectangle, the sides OP and OQ must be perpendicular, and the sides PQ and QD must be equal and parallel.Alternatively, since O and D are opposite corners, the other two vertices P and Q must lie on the sides of ABCD such that OPQD forms a rectangle.Let me consider the coordinates of P and Q.Since P and Q lie on the perimeter of ABCD, their coordinates must satisfy either x=0, x=a, y=0, or y=b.Let me assume that P lies on side AB (from A(0,b) to B(0,0)) and Q lies on side BC (from B(0,0) to C(a,0)). Alternatively, P could lie on AD (from A(0,b) to D(a,b)) and Q on CD (from C(a,0) to D(a,b)). Hmm, I need to figure out where exactly P and Q lie.Wait, since O is inside the original rectangle, and D is at (a,b), the new rectangle OPQD must have P and Q on different sides of ABCD.Let me try to parameterize the coordinates of P and Q.Let me suppose that P lies on side AB, which is the line x=0 from (0,0) to (0,b). So, P would have coordinates (0, p) where 0 ≤ p ≤ b.Similarly, Q lies on side BC, which is the line y=0 from (0,0) to (a,0). So, Q would have coordinates (q, 0) where 0 ≤ q ≤ a.Now, since OPQD is a rectangle, the vectors OP and OQ should be perpendicular and equal in length to the sides of the rectangle.Wait, actually, in a rectangle, the sides are perpendicular, so the vectors OP and OQ should be perpendicular.But OP is from O(r, r) to P(0, p), so vector OP is (-r, p - r).Similarly, OQ is from O(r, r) to Q(q, 0), so vector OQ is (q - r, -r).For these vectors to be perpendicular, their dot product should be zero:(-r)(q - r) + (p - r)(-r) = 0Simplify:-r(q - r) - r(p - r) = 0- rq + r² - rp + r² = 0- rq - rp + 2r² = 0Factor out -r:- r(q + p - 2r) = 0Since r ≠ 0 (because the inradius is positive), we have:q + p - 2r = 0So, q + p = 2rThat's one equation.Additionally, since OPQD is a rectangle, the vector PQ should be equal to the vector OD.Wait, no, in a rectangle, the opposite sides are equal and parallel. So, vector OP should be equal to vector QD, and vector OQ should be equal to vector PD.Wait, let me think again.In rectangle OPQD, the sides are OP, PQ, QD, and DO.Wait, actually, the sides are OP, PQ, QD, and DO, but since it's a rectangle, OP is congruent to QD, and PQ is congruent to DO.Wait, maybe it's better to use the properties of rectangles in terms of coordinates.In a rectangle, the coordinates of the vertices satisfy certain conditions. Specifically, the midpoints of the diagonals coincide, and the slopes of adjacent sides are negative reciprocals (if they are not axis-aligned).But perhaps a better approach is to use the fact that in a rectangle, the sides are perpendicular, so the product of their slopes is -1.Alternatively, since we have coordinates, maybe we can set up equations based on the slopes.But this might get messy. Maybe another approach is to consider that in rectangle OPQD, the sides OP and OQ are adjacent sides, so they should be perpendicular.We already have the condition that q + p = 2r from the dot product.Additionally, since OPQD is a rectangle, the vector PQ should be equal to the vector OD.Wait, vector PQ is from P(0, p) to Q(q, 0), so it's (q, -p).Vector OD is from O(r, r) to D(a, b), so it's (a - r, b - r).In a rectangle, PQ should be equal to OD, but actually, no, PQ is one side and OD is a diagonal. Hmm, maybe that's not the right approach.Wait, in a rectangle, the diagonals are equal and bisect each other. So, the midpoint of OP and the midpoint of QD should be the same.Midpoint of OP: ((r + 0)/2, (r + p)/2) = (r/2, (r + p)/2)Midpoint of QD: ((q + a)/2, (0 + b)/2) = ((q + a)/2, b/2)Since these midpoints must be equal:r/2 = (q + a)/2 => r = q + aand(r + p)/2 = b/2 => r + p = bBut earlier, we had q + p = 2r.So, from r = q + a, we can express q = r - a.From r + p = b, we have p = b - r.Substituting q = r - a and p = b - r into q + p = 2r:(r - a) + (b - r) = 2rSimplify:r - a + b - r = 2r(-a + b) = 2rSo, 2r = b - aBut earlier, we have r = (a + b - sqrt(a² + b²))/2So, 2r = a + b - sqrt(a² + b²)Therefore, from both equations:a + b - sqrt(a² + b²) = b - aSimplify:a + b - sqrt(a² + b²) = b - aSubtract b from both sides:a - sqrt(a² + b²) = -aAdd sqrt(a² + b²) to both sides:a = sqrt(a² + b²) - aAdd a to both sides:2a = sqrt(a² + b²)Square both sides:4a² = a² + b²Subtract a²:3a² = b²So, b = a*sqrt(3)But we know that the area of the original rectangle is a*b = 1.So, a*(a*sqrt(3)) = 1 => a²*sqrt(3) = 1 => a² = 1/sqrt(3) => a = 1/(3^(1/4))But let me compute this properly.Wait, a²*sqrt(3) = 1 => a² = 1/sqrt(3) => a = (1/sqrt(3))^(1/2) = 1/(3^(1/4)).But maybe it's better to express in terms of sqrt(3).Alternatively, since 3a² = b², and a*b = 1, let me solve for a and b.From 3a² = b², we have b = a*sqrt(3)Substitute into a*b = 1:a*(a*sqrt(3)) = 1 => a²*sqrt(3) = 1 => a² = 1/sqrt(3) => a = 1/(3^(1/4)).But 3^(1/4) is the fourth root of 3, which is approximately 1.316, but maybe we can express it in terms of sqrt(3).Wait, 3^(1/4) = sqrt(sqrt(3)).Alternatively, maybe we can rationalize it.But perhaps it's better to keep it as a = 1/(3^(1/4)) and b = sqrt(3)/(3^(1/4)).But let me see if I can express this differently.Wait, let me go back to the equation 2r = b - a.We have r = (a + b - sqrt(a² + b²))/2So, 2r = a + b - sqrt(a² + b²) = b - aTherefore, a + b - sqrt(a² + b²) = b - aSimplify:a + b - sqrt(a² + b²) = b - aSubtract b from both sides:a - sqrt(a² + b²) = -aAdd sqrt(a² + b²) to both sides:a = sqrt(a² + b²) - aAdd a to both sides:2a = sqrt(a² + b²)Square both sides:4a² = a² + b²So, 3a² = b² => b = a*sqrt(3)So, that's consistent.Therefore, the sides of the original rectangle are a and a*sqrt(3), with a*a*sqrt(3) = 1 => a²*sqrt(3) = 1 => a² = 1/sqrt(3) => a = 1/(3^(1/4)).But maybe it's better to express a in terms of sqrt(3).Wait, 3^(1/4) is the same as sqrt(sqrt(3)), so a = 1/sqrt(sqrt(3)) = sqrt(sqrt(3))/sqrt(3) = 3^(1/4)/3^(1/2) = 3^(-1/4).But perhaps it's not necessary to compute the exact value of a and b, since we might be able to find the area of the new rectangle without knowing a and b explicitly.Wait, the area of the new rectangle is the product of the lengths of OP and OQ, since it's a rectangle.But OP and OQ are vectors from O to P and O to Q, which are adjacent sides of the rectangle.But since OP and OQ are perpendicular, the area is |OP| * |OQ|.Alternatively, since we have coordinates, maybe we can compute the vectors OP and OQ, find their lengths, and multiply them.But let's see.From earlier, we have:q + p = 2rAnd from the midpoint condition:r = q + ap = b - rSo, substituting q = r - a and p = b - r into q + p = 2r:(r - a) + (b - r) = 2r => b - a = 2rWhich we already used.So, from b - a = 2r, and r = (a + b - sqrt(a² + b²))/2, we get:b - a = (a + b - sqrt(a² + b²))Which simplifies to 2a = sqrt(a² + b²), as before.So, we have b = a*sqrt(3)Therefore, the original rectangle has sides a and a*sqrt(3), with area a²*sqrt(3) = 1.So, a² = 1/sqrt(3) => a = 1/(3^(1/4)).But let's see if we can find the area of the new rectangle without explicitly finding a and b.Wait, the new rectangle has vertices O(r, r), D(a, b), and the other two vertices P and Q on the sides of the original rectangle.From earlier, we have P(0, p) and Q(q, 0), with p = b - r and q = r - a.So, the coordinates are:O(r, r)P(0, b - r)Q(r - a, 0)D(a, b)Now, to find the area of rectangle OPQD, we can compute the vectors OP and OQ and take their magnitudes, then multiply them since they are perpendicular.Vector OP is from O(r, r) to P(0, b - r): (-r, b - 2r)Vector OQ is from O(r, r) to Q(r - a, 0): (-a, -r)Wait, but earlier we had vector OP as (-r, p - r) = (-r, (b - r) - r) = (-r, b - 2r)Similarly, vector OQ is (q - r, -r) = ((r - a) - r, -r) = (-a, -r)So, vectors OP = (-r, b - 2r) and OQ = (-a, -r)Since OP and OQ are perpendicular, their dot product is zero:(-r)(-a) + (b - 2r)(-r) = 0ra - r(b - 2r) = 0ra - rb + 2r² = 0r(a - b + 2r) = 0Since r ≠ 0, we have a - b + 2r = 0 => a - b = -2rBut from earlier, we have b - a = 2r, which is consistent.So, now, to find the area of the rectangle, we can compute the magnitude of OP times the magnitude of OQ.|OP| = sqrt((-r)^2 + (b - 2r)^2) = sqrt(r² + (b - 2r)^2)|OQ| = sqrt((-a)^2 + (-r)^2) = sqrt(a² + r²)So, area = |OP| * |OQ| = sqrt(r² + (b - 2r)^2) * sqrt(a² + r²)But this seems complicated. Maybe there's a simpler way.Alternatively, since the new rectangle has vertices O, P, Q, D, and we know the coordinates of all four points, we can compute the area using the shoelace formula.The shoelace formula for the area of a polygon given its vertices is:Area = 1/2 |sum_{i=1 to n} (x_i y_{i+1} - x_{i+1} y_i)|Where the vertices are ordered sequentially.So, let's list the coordinates of the new rectangle OPQD in order:O(r, r)P(0, b - r)Q(r - a, 0)D(a, b)Back to O(r, r)So, applying the shoelace formula:Area = 1/2 | (r*(b - r) + 0*0 + (r - a)*b + a*r) - (r*0 + (b - r)*(r - a) + 0*a + b*r) |Let me compute each term step by step.First, compute the terms x_i y_{i+1}:1. O to P: r*(b - r)2. P to Q: 0*0 = 03. Q to D: (r - a)*b4. D to O: a*rSum of x_i y_{i+1}: r(b - r) + 0 + b(r - a) + a r = r b - r² + b r - a b + a rSimplify:r b - r² + b r - a b + a r = 2 r b - r² - a b + a rNow, compute the terms y_i x_{i+1}:1. O to P: r*0 = 02. P to Q: (b - r)*(r - a)3. Q to D: 0*a = 04. D to O: b*rSum of y_i x_{i+1}: 0 + (b - r)(r - a) + 0 + b rExpand (b - r)(r - a):= b r - b a - r² + a rSo, sum of y_i x_{i+1}: b r - b a - r² + a r + b r = 2 b r - b a - r² + a rNow, subtract the two sums:[2 r b - r² - a b + a r] - [2 b r - b a - r² + a r] = (2 r b - r² - a b + a r) - 2 b r + b a + r² - a rSimplify term by term:2 r b - 2 b r = 0- r² + r² = 0- a b + b a = 0a r - a r = 0So, the entire expression inside the absolute value is 0.Wait, that can't be right. If the area is zero, that would mean the points are colinear, which they are not.Hmm, I must have made a mistake in applying the shoelace formula.Wait, let me double-check the order of the points. In the shoelace formula, the points must be ordered either clockwise or counterclockwise without crossing.I listed them as O, P, Q, D, O.Let me plot these points:O is inside the rectangle, P is on side AB, Q is on side BC, D is the opposite corner.So, the order O -> P -> Q -> D -> O should form a rectangle.But perhaps the shoelace formula is giving zero because the points are not ordered correctly or because of some calculation error.Let me try again, carefully.Compute x_i y_{i+1}:1. O(r, r) to P(0, b - r): r*(b - r)2. P(0, b - r) to Q(r - a, 0): 0*0 = 03. Q(r - a, 0) to D(a, b): (r - a)*b4. D(a, b) to O(r, r): a*rSum: r(b - r) + 0 + b(r - a) + a r = r b - r² + b r - a b + a r = 2 r b - r² - a b + a rCompute y_i x_{i+1}:1. O(r, r) to P(0, b - r): r*0 = 02. P(0, b - r) to Q(r - a, 0): (b - r)*(r - a)3. Q(r - a, 0) to D(a, b): 0*a = 04. D(a, b) to O(r, r): b*rSum: 0 + (b - r)(r - a) + 0 + b r = (b r - b a - r² + a r) + b r = 2 b r - b a - r² + a rNow, subtract the two sums:[2 r b - r² - a b + a r] - [2 b r - b a - r² + a r] = (2 r b - r² - a b + a r) - 2 b r + b a + r² - a rSimplify:2 r b - 2 b r = 0- r² + r² = 0- a b + b a = 0a r - a r = 0So, indeed, the result is 0. That suggests that the points are colinear, which is not possible since they form a rectangle. Therefore, I must have made a mistake in the order of the points or in the application of the shoelace formula.Alternatively, maybe the shoelace formula isn't the best approach here because the rectangle is not axis-aligned, and the coordinates are not in a simple order.Perhaps a better approach is to compute the vectors OP and OQ, find their lengths, and then compute the area as the product of these lengths since they are perpendicular.From earlier, we have:OP = (-r, b - 2r)OQ = (-a, -r)We know that OP and OQ are perpendicular, so their dot product is zero, which we already used.Now, let's compute |OP| and |OQ|.|OP| = sqrt(r² + (b - 2r)^2)|OQ| = sqrt(a² + r²)So, area = |OP| * |OQ| = sqrt(r² + (b - 2r)^2) * sqrt(a² + r²)But this seems complicated. Maybe we can express this in terms of a and b.Wait, we know that b = a*sqrt(3), and a*b = 1.So, b = a*sqrt(3), and a*(a*sqrt(3)) = 1 => a² = 1/sqrt(3)So, a = 1/(3^(1/4)) and b = sqrt(3)/(3^(1/4))Let me compute r:r = (a + b - sqrt(a² + b²))/2Compute sqrt(a² + b²):a² + b² = a² + 3a² = 4a² = 4*(1/sqrt(3)) = 4/sqrt(3)So, sqrt(a² + b²) = sqrt(4/sqrt(3)) = 2/(3^(1/4))Therefore, r = (a + b - 2/(3^(1/4)))/2But a = 1/(3^(1/4)) and b = sqrt(3)/(3^(1/4)) = 3^(1/2)/(3^(1/4)) = 3^(1/4)So, a + b = 1/(3^(1/4)) + 3^(1/4)Let me compute this:Let me denote t = 3^(1/4), so t^4 = 3Then, a = 1/t, b = tSo, a + b = 1/t + t = (1 + t²)/tSimilarly, sqrt(a² + b²) = 2/tSo, r = ( (1 + t²)/t - 2/t ) / 2 = ( (1 + t² - 2)/t ) / 2 = ( (t² - 1)/t ) / 2 = (t² - 1)/(2t)So, r = (t² - 1)/(2t)Now, let's compute |OP| and |OQ|.First, |OP| = sqrt(r² + (b - 2r)^2)Compute b - 2r:b = t2r = 2*(t² - 1)/(2t) = (t² - 1)/tSo, b - 2r = t - (t² - 1)/t = (t² - (t² - 1))/t = (t² - t² + 1)/t = 1/tTherefore, |OP| = sqrt(r² + (1/t)^2)Similarly, |OQ| = sqrt(a² + r²) = sqrt( (1/t²) + r² )So, let's compute r²:r = (t² - 1)/(2t)r² = (t² - 1)^2/(4t²) = (t^4 - 2t² + 1)/(4t²)But t^4 = 3, so:r² = (3 - 2t² + 1)/(4t²) = (4 - 2t²)/(4t²) = (2 - t²)/(2t²)Now, compute |OP|:|OP| = sqrt(r² + (1/t)^2) = sqrt( (2 - t²)/(2t²) + 1/t² ) = sqrt( (2 - t² + 2)/(2t²) ) = sqrt( (4 - t²)/(2t²) )Similarly, compute |OQ|:|OQ| = sqrt(1/t² + r²) = sqrt(1/t² + (2 - t²)/(2t²)) = sqrt( (2 + 2 - t²)/(2t²) ) = sqrt( (4 - t²)/(2t²) )So, both |OP| and |OQ| are equal to sqrt( (4 - t²)/(2t²) )Therefore, the area of the rectangle is |OP| * |OQ| = ( (4 - t²)/(2t²) )But t^4 = 3, so t² = sqrt(3)Therefore, 4 - t² = 4 - sqrt(3)And 2t² = 2*sqrt(3)So, area = (4 - sqrt(3))/(2*sqrt(3))Simplify:Multiply numerator and denominator by sqrt(3):= ( (4 - sqrt(3)) * sqrt(3) ) / (2*3) = (4 sqrt(3) - 3)/6So, the area is (4 sqrt(3) - 3)/6But let me check if this makes sense.Wait, the original area is 1, and the new area is (4 sqrt(3) - 3)/6 ≈ (6.928 - 3)/6 ≈ 3.928/6 ≈ 0.6547Which is less than 1, which seems reasonable.But let me see if there's a simpler way to express this.Alternatively, since we have t^4 = 3, t² = sqrt(3), so 4 - t² = 4 - sqrt(3), and 2t² = 2 sqrt(3)So, area = (4 - sqrt(3))/(2 sqrt(3)) = (4)/(2 sqrt(3)) - (sqrt(3))/(2 sqrt(3)) = 2/sqrt(3) - 1/2Rationalize 2/sqrt(3):= (2 sqrt(3))/3 - 1/2So, area = (2 sqrt(3))/3 - 1/2But let me compute this:(2 sqrt(3))/3 ≈ 1.15471/2 = 0.5So, 1.1547 - 0.5 ≈ 0.6547, which matches the earlier approximation.But is there a way to express this as a single fraction?Yes:(2 sqrt(3))/3 - 1/2 = (4 sqrt(3) - 3)/6Which is what we had earlier.So, the area of the new rectangle is (4 sqrt(3) - 3)/6But let me see if this can be simplified further or expressed differently.Alternatively, factor out 1/6:= (4 sqrt(3) - 3)/6I think that's as simplified as it gets.But wait, let me check if I made any mistakes in the calculations.From earlier, we had:Area = |OP| * |OQ| = (4 - t²)/(2t²) where t² = sqrt(3)So, 4 - t² = 4 - sqrt(3)And 2t² = 2 sqrt(3)So, area = (4 - sqrt(3))/(2 sqrt(3)) = (4 sqrt(3) - 3)/6Yes, that seems correct.Alternatively, maybe there's a geometric interpretation that can lead to a simpler solution.Wait, considering that the new rectangle has vertices O and D, and the other two vertices on the sides of the original rectangle, perhaps the area can be found using coordinate geometry or by recognizing some symmetry.But given the time I've spent, I think the area is (4 sqrt(3) - 3)/6But let me check if this is correct by considering specific values.Suppose a = 1, then b = sqrt(3), and the area of the original rectangle is a*b = sqrt(3), which is not 1. So, to make a*b = 1, we have a = 1/(3^(1/4)) and b = sqrt(3)/(3^(1/4)) as before.But let me compute the area numerically:a ≈ 1/(1.316) ≈ 0.7598b ≈ 1.732/1.316 ≈ 1.316So, a ≈ 0.7598, b ≈ 1.316Then, r = (a + b - sqrt(a² + b²))/2Compute sqrt(a² + b²):a² ≈ 0.577, b² ≈ 1.732So, a² + b² ≈ 0.577 + 1.732 ≈ 2.309sqrt(2.309) ≈ 1.519So, r ≈ (0.7598 + 1.316 - 1.519)/2 ≈ (2.0758 - 1.519)/2 ≈ 0.5568/2 ≈ 0.2784Now, compute the area of the new rectangle:(4 sqrt(3) - 3)/6 ≈ (6.928 - 3)/6 ≈ 3.928/6 ≈ 0.6547Alternatively, compute |OP| and |OQ|:|OP| = sqrt(r² + (b - 2r)^2) ≈ sqrt(0.2784² + (1.316 - 2*0.2784)^2) ≈ sqrt(0.0775 + (1.316 - 0.5568)^2) ≈ sqrt(0.0775 + (0.7592)^2) ≈ sqrt(0.0775 + 0.5764) ≈ sqrt(0.6539) ≈ 0.8086|OQ| = sqrt(a² + r²) ≈ sqrt(0.577 + 0.0775) ≈ sqrt(0.6545) ≈ 0.8090So, area ≈ 0.8086 * 0.8090 ≈ 0.6545, which matches the earlier calculation.Therefore, the area of the new rectangle is (4 sqrt(3) - 3)/6But let me see if this can be simplified or expressed in a different form.Alternatively, factor out 1/6:= (4 sqrt(3) - 3)/6I think that's the simplest form.So, the area of the new rectangle is (4 sqrt(3) - 3)/6 square units.

💡First, I need to determine the total combined age of all four Harper children based on the given average age.Since the average age is 9 years for four children, the total combined age is 9 multiplied by 4, which equals 36 years.Next, I'll add up the ages of the three younger children: 6 years, 8 years, and 10 years. This gives a total of 24 years.To find the age of the oldest child, I'll subtract the combined age of the younger children from the total combined age of all four children. So, 36 years minus 24 years equals 12 years.Therefore, the oldest child is 12 years old.