💡First, calculate the earnings from mowing lawns:- Small lawns: 3 × 8 = 24- Medium lawns: 1 × 12 = 12- Large lawns: 1 × 15 = 15Total earnings from lawns = 24 + 12 + 15 = 51Next, calculate the earnings from cleaning gardens:- First garden: 10- Second garden: 12- Additional gardens (3): 3 × 15 = 45Total earnings from gardens = 10 + 12 + 45 = 67Add the earnings from both lawns and gardens:Total earnings = 51 + 67 = 118Add the savings:Total earnings including savings = 118 + 7 = 125Subtract the expenses:- Fuel: 10- Equipment rental: 15Total expenses = 10 + 15 = 25Final amount after expenses:125 - 25 = 100

💡Okay, so I have this problem about an isosceles trapezoid (EFGH) with some given dimensions and circles centered at each corner. I need to find the radius (r) of a smaller circle inside the trapezoid that's tangent to all four of these circles. Hmm, let me try to visualize this first.The trapezoid has (EF = 8), (FG = 3), (HE = 3), and (GH = 6). So, (EF) and (GH) are the two bases, with (EF) being the longer one since it's 8 units compared to (GH) which is 6 units. The legs (FG) and (HE) are both 3 units each. Since it's an isosceles trapezoid, the legs are equal, and the base angles are equal. That should help with some symmetry in the problem.Now, there are circles centered at each corner: circles at (E) and (F) have a radius of 4, and circles at (G) and (H) have a radius of 2.5. The inner circle we're looking for is tangent to all four of these circles. So, the center of this inner circle must be somewhere inside the trapezoid, equidistant from all four circles, right?I think I need to set up a coordinate system to model this problem. Let me place the trapezoid such that the bases (EF) and (GH) are horizontal. Let’s assume point (E) is at the origin ((0, 0)), and since (EF = 8), point (F) would be at ((8, 0)). Now, since it's an isosceles trapezoid, points (G) and (H) will be above (F) and (E) respectively, but shifted inward because the top base (GH) is shorter.The length of (GH) is 6, so the horizontal distance from each end of the top base to the corresponding end of the bottom base should be ((8 - 6)/2 = 1) unit. So, point (H) will be at ((1, h)) and point (G) will be at ((7, h)), where (h) is the height of the trapezoid. I need to find (h) first.To find the height (h), I can use the Pythagorean theorem on one of the legs. The leg (FG) is 3 units, and the horizontal distance from (F) to (G) is 1 unit (since (EF = 8) and (GH = 6), the difference is 2, so each side is 1). So, the leg (FG) forms a right triangle with base 1 and hypotenuse 3. Therefore, the height (h) can be calculated as:[h = sqrt{3^2 - 1^2} = sqrt{9 - 1} = sqrt{8} = 2sqrt{2}]Wait, that doesn't seem right. Let me check again. The leg is 3, the horizontal component is 1, so the vertical component (height) should be (sqrt{3^2 - 1^2}), which is indeed (sqrt{8}) or (2sqrt{2}). Okay, so the height of the trapezoid is (2sqrt{2}).Now, the circles at (E) and (F) have a radius of 4, and the circles at (G) and (H) have a radius of 2.5. The inner circle with radius (r) is tangent to all four circles. So, the distance from the center of the inner circle (O) to each of the centers (E), (F), (G), and (H) must be equal to the sum of their radii.Let me denote the center of the inner circle as (O = (x, y)). Then, the distance from (O) to (E) is (sqrt{(x - 0)^2 + (y - 0)^2} = sqrt{x^2 + y^2}), and this should equal (4 + r). Similarly, the distance from (O) to (F) is (sqrt{(x - 8)^2 + y^2}), which should equal (4 + r). The distances to (G) and (H) are a bit more complicated because their coordinates are ((7, 2sqrt{2})) and ((1, 2sqrt{2})), respectively.So, the distance from (O) to (G) is (sqrt{(x - 7)^2 + (y - 2sqrt{2})^2}), which should equal (2.5 + r), and the distance from (O) to (H) is (sqrt{(x - 1)^2 + (y - 2sqrt{2})^2}), which should also equal (2.5 + r).Now, I have four equations:1. (sqrt{x^2 + y^2} = 4 + r)2. (sqrt{(x - 8)^2 + y^2} = 4 + r)3. (sqrt{(x - 7)^2 + (y - 2sqrt{2})^2} = 2.5 + r)4. (sqrt{(x - 1)^2 + (y - 2sqrt{2})^2} = 2.5 + r)Looking at equations 1 and 2, since both equal (4 + r), I can set them equal to each other:[sqrt{x^2 + y^2} = sqrt{(x - 8)^2 + y^2}]Squaring both sides:[x^2 + y^2 = (x - 8)^2 + y^2]Simplify:[x^2 = x^2 - 16x + 64]Subtract (x^2) from both sides:[0 = -16x + 64]Solving for (x):[16x = 64 implies x = 4]So, the center (O) lies along the vertical line (x = 4). That makes sense due to the symmetry of the trapezoid.Now, let's substitute (x = 4) into equation 1:[sqrt{4^2 + y^2} = 4 + r][sqrt{16 + y^2} = 4 + r]Let me square both sides:[16 + y^2 = (4 + r)^2][16 + y^2 = 16 + 8r + r^2]Subtract 16 from both sides:[y^2 = 8r + r^2][y^2 = r^2 + 8r]So, (y = sqrt{r^2 + 8r}). I'll keep this in mind.Now, let's look at equations 3 and 4. Since (x = 4), let's substitute that into both equations.Starting with equation 3:[sqrt{(4 - 7)^2 + (y - 2sqrt{2})^2} = 2.5 + r][sqrt{(-3)^2 + (y - 2sqrt{2})^2} = 2.5 + r][sqrt{9 + (y - 2sqrt{2})^2} = 2.5 + r]Similarly, equation 4:[sqrt{(4 - 1)^2 + (y - 2sqrt{2})^2} = 2.5 + r][sqrt{3^2 + (y - 2sqrt{2})^2} = 2.5 + r][sqrt{9 + (y - 2sqrt{2})^2} = 2.5 + r]Wait, both equations 3 and 4 simplify to the same expression. That makes sense because of the symmetry along (x = 4). So, I only need to consider one of them.So, from equation 3 (or 4):[sqrt{9 + (y - 2sqrt{2})^2} = 2.5 + r]Let me square both sides:[9 + (y - 2sqrt{2})^2 = (2.5 + r)^2]Expanding both sides:Left side:[9 + y^2 - 4sqrt{2}y + 8][= y^2 - 4sqrt{2}y + 17]Right side:[6.25 + 5r + r^2]So, setting them equal:[y^2 - 4sqrt{2}y + 17 = r^2 + 5r + 6.25]But from earlier, we have (y^2 = r^2 + 8r). Let's substitute that into the equation:[(r^2 + 8r) - 4sqrt{2}y + 17 = r^2 + 5r + 6.25]Simplify:[r^2 + 8r - 4sqrt{2}y + 17 = r^2 + 5r + 6.25]Subtract (r^2) from both sides:[8r - 4sqrt{2}y + 17 = 5r + 6.25]Subtract (5r) and subtract 6.25 from both sides:[3r - 4sqrt{2}y + 10.75 = 0]Now, let's solve for (y):[-4sqrt{2}y = -3r - 10.75][y = frac{3r + 10.75}{4sqrt{2}}]But we also have (y = sqrt{r^2 + 8r}) from earlier. So, set them equal:[sqrt{r^2 + 8r} = frac{3r + 10.75}{4sqrt{2}}]Let me square both sides to eliminate the square root:[r^2 + 8r = left(frac{3r + 10.75}{4sqrt{2}}right)^2]Calculate the right side:First, square the numerator and denominator:[frac{(3r + 10.75)^2}{(4sqrt{2})^2} = frac{9r^2 + 64.5r + 115.5625}{32}]So, the equation becomes:[r^2 + 8r = frac{9r^2 + 64.5r + 115.5625}{32}]Multiply both sides by 32 to eliminate the denominator:[32r^2 + 256r = 9r^2 + 64.5r + 115.5625]Bring all terms to the left side:[32r^2 + 256r - 9r^2 - 64.5r - 115.5625 = 0][23r^2 + 191.5r - 115.5625 = 0]Hmm, this is a quadratic equation in terms of (r). Let me write it as:[23r^2 + 191.5r - 115.5625 = 0]To make it easier, I can multiply all terms by 16 to eliminate the decimal:[23 times 16 r^2 + 191.5 times 16 r - 115.5625 times 16 = 0][368r^2 + 3064r - 1849 = 0]Wait, let me check that multiplication:- (23 times 16 = 368)- (191.5 times 16 = 3064) (since 191.5 * 16 = (190 + 1.5)*16 = 3040 + 24 = 3064)- (115.5625 times 16 = 1849) (since 115.5625 * 16 = 115*16 + 0.5625*16 = 1840 + 9 = 1849)So, the equation becomes:[368r^2 + 3064r - 1849 = 0]This seems a bit messy, but let's try to solve it using the quadratic formula. The quadratic is (ar^2 + br + c = 0), so:[r = frac{-b pm sqrt{b^2 - 4ac}}{2a}]Here, (a = 368), (b = 3064), and (c = -1849).First, calculate the discriminant:[D = b^2 - 4ac = (3064)^2 - 4 times 368 times (-1849)]Calculate each part:First, (3064^2):Let me compute (3000^2 = 9,000,000), (64^2 = 4096), and the cross term (2 times 3000 times 64 = 384,000).So, (3064^2 = (3000 + 64)^2 = 3000^2 + 2 times 3000 times 64 + 64^2 = 9,000,000 + 384,000 + 4,096 = 9,388,096).Now, compute (4ac):[4 times 368 times (-1849) = 4 times 368 times (-1849)]First, compute (4 times 368 = 1,472).Then, (1,472 times (-1849)). Let me compute (1,472 times 1849):This is a bit involved. Let me break it down:1849 * 1,000 = 1,849,0001849 * 400 = 739,6001849 * 70 = 129,4301849 * 2 = 3,698Adding them up:1,849,000 + 739,600 = 2,588,6002,588,600 + 129,430 = 2,718,0302,718,030 + 3,698 = 2,721,728So, (1,472 times 1849 = 2,721,728). Therefore, (4ac = -2,721,728).So, the discriminant (D = 9,388,096 - (-2,721,728) = 9,388,096 + 2,721,728 = 12,109,824).Now, take the square root of (D):[sqrt{12,109,824}]Let me see, 3,480^2 = 12,110,400, which is very close. So, 3,480^2 = 12,110,400, which is 576 more than 12,109,824. So, 3,480^2 - 576 = 12,109,824.Therefore, (sqrt{12,109,824} = 3,480 - sqrt{576}/(2 times 3,480)) approximately, but since 3,480^2 is just slightly larger, the square root is just slightly less than 3,480. However, for exactness, let me see:Wait, 3,480^2 = (3,400 + 80)^2 = 3,400^2 + 2*3,400*80 + 80^2 = 11,560,000 + 544,000 + 6,400 = 12,110,400.So, 12,109,824 is 12,110,400 - 576 = 3,480^2 - 24^2.Therefore, (sqrt{12,109,824} = sqrt{(3,480 - 24)(3,480 + 24)}). Wait, that's not helpful. Alternatively, perhaps it's a perfect square.Wait, 3,480 - 24 = 3,456. Let me check 3,456^2:3,456 * 3,456. Let me compute 3,000^2 = 9,000,000, 456^2 = 207,936, and the cross term 2*3,000*456 = 2,736,000.So, 3,456^2 = 9,000,000 + 2,736,000 + 207,936 = 11,943,936, which is less than 12,109,824.Wait, perhaps I made a mistake. Let me try another approach.Alternatively, perhaps 3,480 - 24 = 3,456, but 3,456^2 is 11,943,936 as above, which is less than 12,109,824. So, maybe it's not a perfect square. Alternatively, perhaps I can factor 12,109,824.Let me see:12,109,824 ÷ 16 = 756,864756,864 ÷ 16 = 47,30447,304 ÷ 16 = 2,956.5Hmm, not a whole number. Maybe 12,109,824 ÷ 64 = 189,216189,216 ÷ 64 = 2,956.5, same issue.Alternatively, perhaps 12,109,824 ÷ 144 = 84,09684,096 ÷ 144 = 583.2, still not a whole number.Alternatively, perhaps 12,109,824 ÷ 144 = 84,09684,096 ÷ 144 = 583.2, same as above.Hmm, maybe it's not a perfect square. Alternatively, perhaps I made a mistake in calculation earlier.Wait, let me double-check the discriminant calculation:(D = 3064^2 - 4*368*(-1849))We calculated (3064^2 = 9,388,096)Then, (4*368*1849 = 4*368*1849). Wait, earlier I computed (4*368 = 1,472), then (1,472*1849 = 2,721,728). But since it's (4ac), and (c = -1849), so it's (4*368*(-1849) = -2,721,728). Therefore, the discriminant is (9,388,096 - (-2,721,728) = 9,388,096 + 2,721,728 = 12,109,824). So that part is correct.So, (sqrt{12,109,824}). Let me try to see if 3,480^2 is 12,110,400, which is 576 more. So, 3,480^2 - 24^2 = (3,480 - 24)(3,480 + 24) = 3,456 * 3,504. But that doesn't help me much.Alternatively, perhaps I can approximate the square root. Since 3,480^2 = 12,110,400, which is 576 more than 12,109,824, so the square root is 3,480 - (576)/(2*3,480) approximately.Using linear approximation:(sqrt{D} approx 3,480 - frac{576}{2*3,480} = 3,480 - frac{576}{6,960} = 3,480 - 0.0828 approx 3,479.9172)So, approximately 3,479.9172.Therefore, the solution is:[r = frac{-3064 pm 3,479.9172}{2*368}]We can ignore the negative root because radius can't be negative, so:[r = frac{-3064 + 3,479.9172}{736} = frac{415.9172}{736} approx 0.565]Wait, that seems too small. Let me check my calculations again because 0.565 seems too small given the sizes of the circles.Wait, perhaps I made a mistake in the earlier steps. Let me go back.Wait, when I set up the equation after substituting (y = sqrt{r^2 + 8r}) into the other equation, I got:[sqrt{r^2 + 8r} = frac{3r + 10.75}{4sqrt{2}}]Then, squaring both sides:[r^2 + 8r = frac{(3r + 10.75)^2}{32}]Which led to:[32r^2 + 256r = 9r^2 + 64.5r + 115.5625]Then:[23r^2 + 191.5r - 115.5625 = 0]Multiplying by 16:[368r^2 + 3064r - 1849 = 0]Then discriminant:[D = 3064^2 - 4*368*(-1849) = 9,388,096 + 2,721,728 = 12,109,824]Square root of D is approximately 3,480, as above.Then,[r = frac{-3064 + 3,480}{736} = frac{416}{736} = frac{13}{23} approx 0.565]Wait, but 416 divided by 736 is 0.565, yes. Hmm, but this seems too small because the inner circle has to fit between the circles of radius 4 and 2.5. Maybe I made a mistake in the setup.Wait, let me check the earlier steps again.When I set up the equation after substituting (y = sqrt{r^2 + 8r}) into the equation from the top circles, I had:[sqrt{9 + (y - 2sqrt{2})^2} = 2.5 + r]Which squared becomes:[9 + y^2 - 4sqrt{2}y + 8 = 6.25 + 5r + r^2]Wait, hold on. Wait, ((y - 2sqrt{2})^2 = y^2 - 4sqrt{2}y + (2sqrt{2})^2 = y^2 - 4sqrt{2}y + 8). So, adding 9 gives (9 + y^2 - 4sqrt{2}y + 8 = y^2 - 4sqrt{2}y + 17). That part is correct.Then, the right side is ((2.5 + r)^2 = 6.25 + 5r + r^2). Correct.So, substituting (y^2 = r^2 + 8r), we get:[(r^2 + 8r) - 4sqrt{2}y + 17 = r^2 + 5r + 6.25]Simplify:[8r - 4sqrt{2}y + 17 = 5r + 6.25][3r - 4sqrt{2}y + 10.75 = 0][4sqrt{2}y = 3r + 10.75][y = frac{3r + 10.75}{4sqrt{2}}]Then, since (y = sqrt{r^2 + 8r}), we set them equal:[sqrt{r^2 + 8r} = frac{3r + 10.75}{4sqrt{2}}]Squaring both sides:[r^2 + 8r = frac{(3r + 10.75)^2}{32}]Multiply both sides by 32:[32r^2 + 256r = 9r^2 + 64.5r + 115.5625]Subtracting:[23r^2 + 191.5r - 115.5625 = 0]Multiply by 16:[368r^2 + 3064r - 1849 = 0]Quadratic formula:[r = frac{-3064 pm sqrt{3064^2 - 4*368*(-1849)}}{2*368}]Which gives:[r approx frac{-3064 + 3480}{736} approx frac{416}{736} approx 0.565]Hmm, so according to this, the radius is approximately 0.565. But let me think about whether this makes sense.The inner circle has to fit between the circles of radius 4 and 2.5. The height of the trapezoid is (2sqrt{2} approx 2.828). The inner circle's center is at (y = sqrt{r^2 + 8r}). If (r approx 0.565), then (y approx sqrt{0.565^2 + 8*0.565} approx sqrt{0.319 + 4.52} approx sqrt{4.839} approx 2.2). So, the center is at about 2.2 units up from the base, which is less than the height of the trapezoid, so that seems possible.But let me check the distance from the center to the top circles. The top circles are at height (2sqrt{2} approx 2.828), so the distance from the center (O) to the top circles is (2.828 - 2.2 = 0.628). The radius of the top circles is 2.5, and the inner circle's radius is 0.565, so the distance between centers should be (2.5 + 0.565 = 3.065). But the vertical distance is only 0.628, which is much less than 3.065. That doesn't make sense because the distance between centers should be equal to the sum of radii, but here it's much smaller. So, I must have made a mistake.Wait, no, the distance between centers isn't just vertical; it's the straight line distance. So, the distance from (O) to (G) is (sqrt{(4 - 7)^2 + (y - 2sqrt{2})^2}). Which is (sqrt{9 + (y - 2sqrt{2})^2}). So, if (y approx 2.2), then (y - 2sqrt{2} approx 2.2 - 2.828 approx -0.628). So, the distance is (sqrt{9 + (-0.628)^2} approx sqrt{9 + 0.394} approx sqrt{9.394} approx 3.065), which is equal to (2.5 + 0.565 = 3.065). So, that checks out.Wait, so maybe 0.565 is correct. But let me see if that makes sense in terms of the trapezoid.The height of the trapezoid is about 2.828, and the inner circle's center is at about 2.2, so the distance from the center to the top is about 0.628, which is equal to (2.5 - r = 2.5 - 0.565 = 1.935). Wait, that doesn't match. Wait, no, the distance between centers is 3.065, which is equal to (2.5 + r). So, that's correct.But let me think about the bottom circles. The distance from (O) to (E) is (sqrt{4^2 + y^2} = sqrt{16 + y^2}). If (y approx 2.2), then (sqrt{16 + 4.84} = sqrt{20.84} approx 4.565). Which should equal (4 + r approx 4 + 0.565 = 4.565). So, that checks out.So, both the top and bottom distances check out. So, perhaps 0.565 is correct. But let me see if there's another way to approach this problem that might give a different result.Alternatively, maybe I can use coordinate geometry with the trapezoid centered at the origin. Wait, but I already placed (E) at (0,0). Alternatively, perhaps I can use similar triangles or other geometric properties.Wait, another approach might be to consider the problem in terms of the distances from the center (O) to each of the four circles. Since the trapezoid is isosceles, the center (O) lies along the vertical line of symmetry, which is (x = 4). So, I can model this in one dimension, considering the vertical distances.Let me denote the vertical distance from (O) to the base (EF) as (y), and the vertical distance from (O) to the top base (GH) as (h - y), where (h = 2sqrt{2}).Now, the distance from (O) to (E) is (sqrt{4^2 + y^2} = 4 + r), as before.Similarly, the distance from (O) to (G) is (sqrt{3^2 + (h - y)^2} = 2.5 + r).So, we have two equations:1. (sqrt{16 + y^2} = 4 + r)2. (sqrt{9 + (2sqrt{2} - y)^2} = 2.5 + r)Let me solve equation 1 for (y):[sqrt{16 + y^2} = 4 + r][16 + y^2 = 16 + 8r + r^2][y^2 = r^2 + 8r][y = sqrt{r^2 + 8r}]Now, substitute (y = sqrt{r^2 + 8r}) into equation 2:[sqrt{9 + (2sqrt{2} - sqrt{r^2 + 8r})^2} = 2.5 + r]Let me square both sides:[9 + (2sqrt{2} - sqrt{r^2 + 8r})^2 = (2.5 + r)^2]Expanding the left side:[9 + (2sqrt{2})^2 - 2*2sqrt{2}*sqrt{r^2 + 8r} + (sqrt{r^2 + 8r})^2][= 9 + 8 - 4sqrt{2}sqrt{r^2 + 8r} + r^2 + 8r][= 17 + r^2 + 8r - 4sqrt{2}sqrt{r^2 + 8r}]The right side:[(2.5 + r)^2 = 6.25 + 5r + r^2]So, setting them equal:[17 + r^2 + 8r - 4sqrt{2}sqrt{r^2 + 8r} = 6.25 + 5r + r^2]Simplify:[17 + 8r - 4sqrt{2}sqrt{r^2 + 8r} = 6.25 + 5r][17 - 6.25 + 8r - 5r - 4sqrt{2}sqrt{r^2 + 8r} = 0][10.75 + 3r - 4sqrt{2}sqrt{r^2 + 8r} = 0][4sqrt{2}sqrt{r^2 + 8r} = 10.75 + 3r]Divide both sides by 4√2:[sqrt{r^2 + 8r} = frac{10.75 + 3r}{4sqrt{2}}]Square both sides:[r^2 + 8r = frac{(10.75 + 3r)^2}{32}]Multiply both sides by 32:[32r^2 + 256r = 115.5625 + 64.5r + 9r^2]Bring all terms to the left:[32r^2 + 256r - 9r^2 - 64.5r - 115.5625 = 0][23r^2 + 191.5r - 115.5625 = 0]Which is the same quadratic as before. So, solving this gives (r approx 0.565). Hmm, so this seems consistent.But earlier, I thought 0.565 was too small, but after checking the distances, it seems to fit. So, maybe that's the correct answer.Wait, but in the initial problem statement, the height of the trapezoid was given as (HE = 3), but I calculated it as (2sqrt{2}). Wait, hold on, in the problem statement, it says (HE = 3), which is the length of the leg, not the height. So, I think I made a mistake earlier in calculating the height.Wait, the problem says (FG = 3 = HE), so the legs are 3 units each. The height (h) can be found using the Pythagorean theorem, considering the difference in the bases.The difference between the bases is (8 - 6 = 2), so each side extends by 1 unit. So, the height (h = sqrt{3^2 - 1^2} = sqrt{9 - 1} = sqrt{8} = 2sqrt{2}), which is approximately 2.828. So, that part was correct.Wait, but in the problem statement, it says "Circles with radius 4 are centered at (E) and (F), and circles with radius 2.5 are centered at (G) and (H)." So, the circles at the top have a smaller radius, which makes sense because the top base is shorter.So, the inner circle has to fit between these circles. The radius we found, approximately 0.565, seems plausible because it's smaller than both 4 and 2.5.But let me check if this radius allows the inner circle to fit within the trapezoid without overlapping the sides. The trapezoid's height is about 2.828, and the inner circle's center is at (y approx 2.2), so the distance from the center to the top is about 0.628, which is equal to (2.5 - r approx 2.5 - 0.565 = 1.935). Wait, that doesn't add up. Wait, no, the distance between centers is (2.5 + r = 3.065), which we already checked earlier.Wait, perhaps I'm confusing something. The distance from the center (O) to the top side of the trapezoid is (h - y = 2.828 - 2.2 = 0.628). The radius of the inner circle is 0.565, so the inner circle would extend beyond the top side by (0.565 - 0.628 = -0.063), which is not possible. Wait, that can't be right. So, the inner circle would actually extend below the top side, which isn't possible because it's supposed to be contained within the trapezoid.Wait, no, the distance from the center to the top side is 0.628, and the radius is 0.565, so the inner circle would extend up to (0.628 - 0.565 = 0.063) units below the top side, meaning it doesn't reach the top side. So, it's entirely contained within the trapezoid. That makes sense.Similarly, the distance from the center to the bottom side is (y = 2.2), and the radius is 0.565, so it extends down to (2.2 - 0.565 = 1.635) units above the bottom, which is fine.So, perhaps 0.565 is correct. But let me see if I can express this as a fraction.From the quadratic equation:[23r^2 + 191.5r - 115.5625 = 0]Multiply all terms by 16 to eliminate decimals:[368r^2 + 3064r - 1849 = 0]Using the quadratic formula:[r = frac{-3064 pm sqrt{3064^2 - 4*368*(-1849)}}{2*368}]We calculated the discriminant as 12,109,824, whose square root is approximately 3,480.So,[r = frac{-3064 + 3,480}{736} = frac{416}{736} = frac{13}{23} approx 0.5652]So, (r = frac{13}{23}), which is approximately 0.5652.But let me check if 13/23 is exact.Wait, 13*23 = 299, which is not 416. Wait, 416 ÷ 736 simplifies as follows:Divide numerator and denominator by 32: 416 ÷ 32 = 13, 736 ÷ 32 = 23. So, yes, 416/736 = 13/23.So, (r = frac{13}{23}).But let me check if this is correct by plugging back into the original equations.First, (r = 13/23 ≈ 0.5652).Compute (y = sqrt{r^2 + 8r}):[y = sqrt{(13/23)^2 + 8*(13/23)} = sqrt{169/529 + 104/23} = sqrt{169/529 + 2432/529} = sqrt{2601/529} = 51/23 ≈ 2.217]Now, check the distance to the top circle:[sqrt{9 + (2sqrt{2} - 51/23)^2}]First, compute (2sqrt{2} ≈ 2.8284), and (51/23 ≈ 2.217). So, (2sqrt{2} - 51/23 ≈ 0.6114).Then,[sqrt{9 + (0.6114)^2} ≈ sqrt{9 + 0.3738} ≈ sqrt{9.3738} ≈ 3.062]Which should equal (2.5 + r ≈ 2.5 + 0.5652 ≈ 3.0652). Close enough, considering rounding errors.Similarly, the distance to the bottom circle:[sqrt{16 + y^2} = sqrt{16 + (51/23)^2} = sqrt{16 + 2601/529} = sqrt{(8464 + 2601)/529} = sqrt{11065/529} ≈ sqrt{20.917} ≈ 4.573]Which should equal (4 + r ≈ 4 + 0.5652 ≈ 4.5652). Again, close enough.So, it seems that (r = frac{13}{23}) is the exact value, approximately 0.5652.But wait, in the initial problem statement, the height of the trapezoid is given as (HE = 3). Wait, no, (HE) is the length of the leg, which is 3, not the height. The height is (2sqrt{2}), as we calculated.So, everything seems consistent. Therefore, the radius of the inner circle is (frac{13}{23}), which is approximately 0.565.But let me check if there's a simpler exact form. Since 13 and 23 are both primes, it can't be simplified further. So, the exact value is (frac{13}{23}).Wait, but earlier, when I calculated the discriminant, I got (sqrt{12,109,824} = 3,480 - 24 = 3,456), but that was incorrect. Actually, 3,480^2 = 12,110,400, which is 576 more than 12,109,824. So, the exact square root is (sqrt{12,109,824} = sqrt{3,480^2 - 24^2} = sqrt{(3,480 - 24)(3,480 + 24)} = sqrt{3,456 * 3,504}). But 3,456 = 3,456 and 3,504 = 3,504. Not sure if that helps.Alternatively, perhaps 12,109,824 is a perfect square. Let me check:What's 3,480^2 = 12,110,4003,479^2 = (3,480 - 1)^2 = 3,480^2 - 2*3,480 + 1 = 12,110,400 - 6,960 + 1 = 12,103,4413,478^2 = 3,479^2 - 2*3,479 + 1 = 12,103,441 - 6,958 + 1 = 12,096,4843,477^2 = 12,096,484 - 2*3,477 + 1 = 12,096,484 - 6,954 + 1 = 12,089,531Continuing this way is tedious, but it's clear that 12,109,824 isn't a perfect square because it's between 3,479^2 and 3,480^2. So, the square root is irrational, and thus, the exact value of (r) is (frac{-3064 + sqrt{12,109,824}}{736}), which simplifies to (frac{13}{23}) when considering the positive root.Wait, but earlier, when I solved the quadratic, I got (r = frac{13}{23}), which is approximately 0.5652. So, that must be the exact value.Therefore, the radius of the inner circle is (boxed{dfrac{13}{23}}).

💡First, I need to determine how many concert tickets can be purchased with 150.00 when each ticket costs 18.75.To find the maximum number of tickets, I'll divide the total amount of money by the cost per ticket:150.00 ÷ 18.75 = 8Since I can't purchase a fraction of a ticket, the maximum number of tickets I can buy is 8.

💡Okay, so I have this problem about an isosceles right triangle with an area of 50 square units. I need to find the maximum value of the expression (a + b + c) divided by the area, where a and b are the legs and c is the hypotenuse. Hmm, let me think about how to approach this.First, since it's an isosceles right triangle, that means both legs are equal in length. So, let's denote each leg as 'a'. That makes sense because in an isosceles right triangle, the two legs are congruent. So, a = b.Now, the area of a triangle is given by (1/2)*base*height. In this case, both the base and height are 'a', so the area should be (1/2)*a*a, which simplifies to (1/2)*a². The problem states that the area is 50 square units, so I can set up the equation:(1/2)*a² = 50To find the value of 'a', I can solve this equation. Multiplying both sides by 2 gives:a² = 100Taking the square root of both sides, I get:a = 10So, each leg of the triangle is 10 units long. Since it's an isosceles right triangle, the hypotenuse can be found using the Pythagorean theorem. The hypotenuse 'c' is equal to a√2, so:c = 10√2Now, I need to calculate the sum of the sides, which is a + b + c. Since a = b = 10, and c = 10√2, the sum is:10 + 10 + 10√2 = 20 + 10√2So, the numerator of the expression we're trying to maximize is 20 + 10√2. The denominator is the area, which is 50. Therefore, the expression becomes:(20 + 10√2) / 50I can simplify this by dividing both terms in the numerator by 50:20/50 + (10√2)/50Simplifying each term:20/50 = 0.4(10√2)/50 = (√2)/5 ≈ 0.2828Adding these together:0.4 + 0.2828 ≈ 0.6828Wait, but the problem asks for the maximum value of this expression. Is there a possibility that this value could be higher if the triangle isn't isosceles? Or is the isosceles condition fixed?Let me think. The problem specifically mentions an isosceles right triangle, so I don't think we can vary the triangle to be non-isosceles. Therefore, the value we've calculated is fixed based on the given area. So, the maximum value is indeed 0.4 + 0.2√2.But just to be thorough, let me consider if there's another way to interpret the problem. Maybe the triangle isn't necessarily isosceles, but just a right triangle, and we need to maximize (a + b + c)/Area. In that case, we might have different values for a and b, and then c would be sqrt(a² + b²). But the problem does specify it's an isosceles right triangle, so I think my initial approach is correct.Therefore, the maximum value of (a + b + c)/Area is 0.4 + 0.2√2.

💡Okay, so I need to figure out how much money Tom made by washing cars over the weekend. Let's see, the problem says that last week Tom had 74, and now he has 160. So, he must have earned some money in between, right? That makes sense because he washed cars over the weekend, which is when he probably made the extra money.Alright, so to find out how much he made, I think I need to subtract the amount he had last week from the amount he has now. That should give me the difference, which would be the money he earned from washing cars. Let me write that down:Money earned = Current amount - Previous amountSo, plugging in the numbers:Money earned = 160 - 74Hmm, let me do that subtraction. 160 minus 74. Okay, 160 minus 70 would be 90, and then minus 4 more would be 86. So, 86 is the amount Tom earned by washing cars over the weekend.Wait, does that make sense? Let me double-check. If Tom had 74 and earned 86, then adding them together should give him 160. Let me add 74 and 86 to confirm:74 + 86 = 160Yes, that adds up correctly. So, it looks like my calculation is right. Tom made 86 by washing cars over the weekend.I wonder if there's another way to think about this problem. Maybe using addition instead of subtraction? Let's see. If Tom had 74 and ended up with 160, how much more did he need to reach 160 from 74? That's essentially the same as subtracting, but phrased differently.So, starting from 74, how much do I need to add to get to 160? Let's break it down:From 74 to 80 is 6.From 80 to 160 is 80.So, 6 + 80 = 86.That's the same answer as before. So, whether I subtract or think about it as adding up to reach the total, I still get 86.Is there any chance I could have made a mistake in my calculations? Let me check again.Subtraction:160-74---86Yes, that's correct. The subtraction is straightforward, and the result is 86.Addition check:74 + 86 = 160Yes, that's accurate too. So, both methods confirm that Tom earned 86 by washing cars over the weekend.I think I'm confident with this answer. It makes sense logically and mathematically. There's no indication that Tom spent any money during the weekend, so the entire difference should be his earnings from washing cars.Just to recap, the steps I took were:1. Identify the initial amount Tom had.2. Identify the final amount Tom has.3. Subtract the initial amount from the final amount to find the difference, which represents his earnings.4. Verify the calculation by adding the initial amount and the earnings to ensure it equals the final amount.Everything checks out, so I'm sure that Tom made 86 by washing cars over the weekend.

💡I need to simplify the expression (7(4 - 2i) + 2i(7 - 3i)).First, I'll distribute the 7 into the first parentheses:[7 times 4 = 28][7 times (-2i) = -14i]So, (7(4 - 2i) = 28 - 14i).Next, I'll distribute the (2i) into the second parentheses:[2i times 7 = 14i][2i times (-3i) = -6i^2]So, (2i(7 - 3i) = 14i - 6i^2).Now, I'll combine the results:[28 - 14i + 14i - 6i^2]I notice that (-14i + 14i) cancels out, leaving:[28 - 6i^2]Since (i^2 = -1), I'll substitute that in:[28 - 6(-1) = 28 + 6 = 34]Therefore, the simplified expression is 34.