💡Alright, so I came across this interesting geometry problem today. It goes like this: "Can a cube with dimensions 20 × 20 × 20 be cut into rectangular parallelepipeds such that the length, width, and height of each parallelepiped are consecutive natural numbers in some order?" Hmm, okay, let's try to unpack this step by step.First off, I need to make sure I understand the terms here. A rectangular parallelepiped is just a 3D shape with all faces being rectangles, right? So, it's like a box, but not necessarily a cube. The key here is that each of these boxes must have dimensions that are consecutive natural numbers. That means for each box, if one side is, say, 5 units, the other sides must be 4 and 6 units, or some permutation of that.Alright, so the cube we're dealing with is 20 units on each side. The question is whether we can divide this cube into smaller boxes where each box has sides of three consecutive numbers. Let's think about what that implies.First, let's consider the volume of the cube. The volume of a cube is side length cubed, so in this case, it's 20 × 20 × 20, which is 8000 cubic units. Now, if we're going to divide this cube into smaller boxes, the sum of the volumes of these boxes must add up to 8000.Each smaller box has dimensions that are three consecutive natural numbers. Let's denote these numbers as n, n+1, and n+2. So, the volume of each such box would be n × (n+1) × (n+2). Now, here's an interesting point: among any three consecutive natural numbers, one of them must be divisible by 3. That's because every third number is a multiple of 3. So, for example, if n is divisible by 3, then n is a multiple of 3. If n leaves a remainder of 1 when divided by 3, then n+2 is divisible by 3. Similarly, if n leaves a remainder of 2 when divided by 3, then n+1 is divisible by 3. This means that the volume of each box, which is n × (n+1) × (n+2), is always divisible by 3. Therefore, each box's volume is a multiple of 3. Now, if we're summing up these volumes to get the total volume of the cube, which is 8000, we need to check if 8000 is divisible by 3. Let's do that. To check if a number is divisible by 3, we can add up its digits and see if the sum is divisible by 3. For 8000, the digits are 8, 0, 0, 0. Adding them up gives 8, which is not divisible by 3. Therefore, 8000 is not divisible by 3.But wait a minute, if each box's volume is divisible by 3, then the total volume should also be divisible by 3, right? Because adding up multiples of 3 should give another multiple of 3. But we just saw that 8000 isn't divisible by 3. That seems like a contradiction.So, this suggests that it's impossible to divide the cube into such boxes because the total volume doesn't align with the divisibility requirement imposed by the dimensions of the smaller boxes. Let me double-check my reasoning to make sure I haven't missed anything. 1. The cube's volume is 8000, which isn't divisible by 3.2. Each smaller box has a volume that is a product of three consecutive numbers, which must be divisible by 3.3. Therefore, the sum of these volumes (which is 8000) must be divisible by 3.4. Since 8000 isn't divisible by 3, this leads to a contradiction.Yes, that seems solid. I don't see any flaws in this logic. It all comes down to the fact that the total volume isn't compatible with the required divisibility of the smaller boxes' volumes.I guess another way to think about it is modular arithmetic. If we consider everything modulo 3, the total volume should be congruent to 0 modulo 3 because each smaller volume is 0 modulo 3. But 8000 modulo 3 is 8000 ÷ 3, which is 2666 with a remainder of 2. So, 8000 ≡ 2 mod 3, which isn't 0. Therefore, it's impossible.Is there any other angle to approach this from? Maybe considering the possible dimensions of the smaller boxes? Let's see. Suppose we try to find a set of three consecutive numbers whose product divides 8000. Let's see what possible values of n could be.Starting with small n:- n=1: 1×2×3=6- n=2: 2×3×4=24- n=3: 3×4×5=60- n=4: 4×5×6=120- n=5: 5×6×7=210- n=6: 6×7×8=336- n=7: 7×8×9=504- n=8: 8×9×10=720- n=9: 9×10×11=990- n=10: 10×11×12=1320- n=11: 11×12×13=1716- n=12: 12×13×14=2184- n=13: 13×14×15=2730- n=14: 14×15×16=3360- n=15: 15×16×17=4080- n=16: 16×17×18=4896- n=17: 17×18×19=5814- n=18: 18×19×20=6840- n=19: 19×20×21=7980- n=20: 20×21×22=9240Wait, n=20 gives a volume larger than 8000, so that's too big. Looking through these, none of these products divide 8000 evenly. For example, 6 divides 8000? 8000 ÷ 6 is approximately 1333.33, which isn't an integer. 24 divides 8000? 8000 ÷ 24 is approximately 333.33, still not an integer. 60 divides 8000? 8000 ÷ 60 is approximately 133.33, again not an integer. Continuing this way, none of these volumes divide 8000 evenly. So, even if we tried to use these smaller boxes, we couldn't fill up the cube completely without leaving gaps or overlapping, which isn't allowed.This further reinforces the idea that it's impossible to partition the cube as described.Another thought: maybe using different sizes of these boxes? Like, mixing different n values to make up the total volume. But even if we do that, the total volume would still need to be a multiple of 3, which it isn't. So, regardless of how we combine these boxes, their total volume would always be a multiple of 3, but 8000 isn't. Therefore, it's impossible.Is there any way around this? Maybe if the dimensions aren't strictly consecutive? But the problem specifically says they must be consecutive natural numbers in some order. So, we can't have non-consecutive dimensions.What if we consider that the order doesn't matter? Like, for each box, the dimensions could be arranged in any order, but they still have to be consecutive numbers. But that doesn't change the fact that each box's volume is divisible by 3, so the total volume still needs to be divisible by 3.Hmm, I don't see a way out of this. It seems like the key issue is the total volume not being divisible by 3, while each smaller box's volume is. Therefore, it's impossible to partition the cube in the way described.Just to wrap up, let's recap:- Cube volume: 8000, not divisible by 3.- Each smaller box's volume: divisible by 3.- Therefore, sum of smaller boxes' volumes: divisible by 3.- But 8000 isn't divisible by 3.- Contradiction.- Conclusion: Impossible.Yeah, that seems to cover it. I don't think I've missed anything here. The crux is the divisibility by 3, which makes it impossible to partition the cube as required.

💡Okay, so I have this problem here: Given that (2 - x)^2015 is expanded into a polynomial a0 + a1x + a2x² + ... + a2015x^2015, I need to find the value of the ratio (a0 + a2 + a4 + ... + a2014) divided by (a1 + a3 + a5 + ... + a2015). Hmm, that seems a bit intimidating because of the high exponent, but maybe there's a clever way to approach it without expanding everything.First, I remember that when dealing with polynomials, plugging in specific values for x can help extract information about the coefficients. For example, plugging in x = 1 gives the sum of all coefficients, and plugging in x = -1 gives an alternating sum of the coefficients. Maybe I can use these two pieces of information to find the sums of the even and odd indexed coefficients separately.Let me write down what I know:(2 - x)^2015 = a0 + a1x + a2x² + ... + a2015x^2015.If I set x = 1, the left side becomes (2 - 1)^2015 = 1^2015 = 1. On the right side, plugging in x = 1 gives a0 + a1 + a2 + ... + a2015. So, equation (1) is:a0 + a1 + a2 + ... + a2015 = 1.Now, if I set x = -1, the left side becomes (2 - (-1))^2015 = (2 + 1)^2015 = 3^2015. On the right side, plugging in x = -1 gives a0 - a1 + a2 - a3 + ... + a2014 - a2015. So, equation (2) is:a0 - a1 + a2 - a3 + ... + a2014 - a2015 = 3^2015.Now, I have two equations:1. Sum of all coefficients: S = a0 + a1 + a2 + ... + a2015 = 1.2. Alternating sum: T = a0 - a1 + a2 - a3 + ... + a2014 - a2015 = 3^2015.I need to find the sum of the even-indexed coefficients, which is a0 + a2 + a4 + ... + a2014, and the sum of the odd-indexed coefficients, which is a1 + a3 + a5 + ... + a2015. Let me denote these as E and O respectively.So, E = a0 + a2 + a4 + ... + a2014,and O = a1 + a3 + a5 + ... + a2015.From equation (1), S = E + O = 1.From equation (2), T = E - O = 3^2015.Now, I have a system of two equations:1. E + O = 12. E - O = 3^2015I can solve this system for E and O. Let's add the two equations:(E + O) + (E - O) = 1 + 3^20152E = 1 + 3^2015So, E = (1 + 3^2015)/2.Similarly, subtracting the second equation from the first:(E + O) - (E - O) = 1 - 3^20152O = 1 - 3^2015So, O = (1 - 3^2015)/2.Now, the ratio we're supposed to find is E/O. Let's compute that:E/O = [(1 + 3^2015)/2] / [(1 - 3^2015)/2].The denominators 2 cancel out, so we have:E/O = (1 + 3^2015)/(1 - 3^2015).That simplifies to (1 + 3^2015)/(1 - 3^2015).Wait, let me double-check my steps to make sure I didn't make a mistake.1. Plugging x = 1 gives S = 1.2. Plugging x = -1 gives T = 3^2015.3. S = E + O, T = E - O.4. Solving for E and O gives E = (S + T)/2 = (1 + 3^2015)/2 and O = (S - T)/2 = (1 - 3^2015)/2.5. Then, E/O is (1 + 3^2015)/(1 - 3^2015).Yes, that seems correct. I don't see any errors in the reasoning. So, the ratio is (1 + 3^2015)/(1 - 3^2015).I wonder if there's another way to approach this problem, maybe using generating functions or binomial coefficients directly. Let me think about it.The polynomial is (2 - x)^2015. The coefficients a_k can be expressed using the binomial theorem:a_k = C(2015, k) * 2^(2015 - k) * (-1)^k.So, the sum of the even-indexed coefficients E is the sum over k even of C(2015, k) * 2^(2015 - k) * (-1)^k.Similarly, the sum of the odd-indexed coefficients O is the sum over k odd of C(2015, k) * 2^(2015 - k) * (-1)^k.But calculating these sums directly seems complicated. Instead, using the substitution method with x = 1 and x = -1 is much more straightforward and avoids dealing with each term individually.Another thought: since the polynomial is (2 - x)^2015, which is symmetric in a certain way, but I don't think symmetry helps here because the coefficients are not symmetric. The coefficients are determined by the binomial expansion, which isn't symmetric unless the polynomial is a palindrome, which (2 - x)^2015 isn't.Therefore, the substitution method is the most efficient way to find E and O.Just to make sure, let me compute E and O for a smaller exponent to see if the method works.Suppose instead of 2015, we have (2 - x)^3.Expanding this, we get 8 - 12x + 6x² - x³.So, a0 = 8, a1 = -12, a2 = 6, a3 = -1.Then, E = a0 + a2 = 8 + 6 = 14,O = a1 + a3 = -12 + (-1) = -13.Now, let's use the substitution method:Set x = 1: (2 - 1)^3 = 1 = 8 - 12 + 6 - 1 = 1. Correct.Set x = -1: (2 - (-1))^3 = 3^3 = 27 = 8 + 12 + 6 + 1 = 27. Correct.Then, E = (1 + 27)/2 = 14,O = (1 - 27)/2 = -13.So, the ratio E/O = 14 / (-13) = -14/13.Which matches our manual calculation. So, the substitution method works for a smaller exponent, which gives me confidence that it's correct for the larger exponent as well.Therefore, I can be confident that the ratio is indeed (1 + 3^2015)/(1 - 3^2015).**Final Answer**The value of the ratio is boxed{dfrac{1 + 3^{2015}}{1 - 3^{2015}}}.

💡Okay, let's see. I have this problem where I'm given that a and b are positive numbers, and there's this logarithmic equation: log base 4 of (1/a + 1/b) equals log base 2 of the square root of (1/(ab)). I need to find the minimum value of (1/a + 1/b). The options are A: 8, B:4, C:1, D:1/4.Hmm, logarithms can sometimes be tricky, but maybe I can simplify this equation by converting the logs to the same base or something. Let me recall that log base 4 is the same as log base 2 squared, so maybe I can express log base 4 in terms of log base 2.I remember that log base a of b is equal to (log base c of b) divided by (log base c of a). So, log base 4 of x is equal to (log base 2 of x) divided by (log base 2 of 4). Since log base 2 of 4 is 2, that means log base 4 of x is (1/2) log base 2 of x. So, applying that to the left side of the equation:log₄(1/a + 1/b) = (1/2) log₂(1/a + 1/b)And the right side is log₂√(1/(ab)). Now, the square root is the same as raising to the power of 1/2, so log₂(√(1/(ab))) is equal to (1/2) log₂(1/(ab)).So now, the equation becomes:(1/2) log₂(1/a + 1/b) = (1/2) log₂(1/(ab))Since both sides have a factor of 1/2, I can multiply both sides by 2 to eliminate them:log₂(1/a + 1/b) = log₂(1/(ab))Now, since the logs are equal and they have the same base, their arguments must be equal. So:1/a + 1/b = 1/(ab)Hmm, okay. Let me write that down:1/a + 1/b = 1/(ab)I can combine the left side over a common denominator:(b + a)/(ab) = 1/(ab)So, (a + b)/ab = 1/abSince ab is positive (because a and b are positive), I can multiply both sides by ab:a + b = 1Oh, interesting. So, a + b equals 1. So, now I have that a + b = 1, and I need to find the minimum value of 1/a + 1/b.Wait, but 1/a + 1/b is the same as (a + b)/(ab), right? Because 1/a + 1/b = (b + a)/(ab). So, since a + b = 1, that means:1/a + 1/b = 1/(ab)So, now I need to find the minimum value of 1/(ab). But since a and b are positive and a + b = 1, I can use the AM-GM inequality to find the maximum value of ab, which would then give me the minimum value of 1/(ab).I remember that for positive numbers a and b, the arithmetic mean is greater than or equal to the geometric mean. So, (a + b)/2 >= sqrt(ab). Since a + b = 1, that becomes:1/2 >= sqrt(ab)Squaring both sides:1/4 >= abSo, ab <= 1/4. Therefore, the maximum value of ab is 1/4, which occurs when a = b = 1/2.Therefore, the minimum value of 1/(ab) is 1/(1/4) = 4.So, the minimum value of 1/a + 1/b is 4.Let me double-check that. If a = b = 1/2, then 1/a + 1/b = 2 + 2 = 4. And indeed, a + b = 1/2 + 1/2 = 1, which satisfies the earlier condition. So, that seems correct.I think that makes sense. So, the answer should be B:4.**Final Answer**boxed{B}

💡Okay, so I have this quadratic equation: (x^2 + tx + 2 = 0), where (t) is a real number. The roots of this equation are (x_1) and (x_2), and I know that the absolute difference between the roots is (2sqrt{2}). I need to find the possible values of (t) from the given options: A: -4, B: -2, C: 0, D: 4.First, I remember that for any quadratic equation (ax^2 + bx + c = 0), the sum of the roots is (-b/a) and the product is (c/a). In this case, since the equation is (x^2 + tx + 2 = 0), the sum of the roots (x_1 + x_2 = -t) and the product (x_1 x_2 = 2).Now, the problem gives me (|x_1 - x_2| = 2sqrt{2}). I need to relate this to the coefficients of the equation. I recall that the difference of the roots can be found using the formula (sqrt{(x_1 + x_2)^2 - 4x_1 x_2}). This comes from expanding ((x_1 - x_2)^2 = (x_1 + x_2)^2 - 4x_1 x_2).So, let's compute ((x_1 - x_2)^2):[(x_1 - x_2)^2 = (x_1 + x_2)^2 - 4x_1 x_2 = (-t)^2 - 4 times 2 = t^2 - 8]Given that (|x_1 - x_2| = 2sqrt{2}), squaring both sides gives:[(2sqrt{2})^2 = 8 = t^2 - 8]So, solving for (t^2):[t^2 = 8 + 8 = 16]Taking the square root of both sides:[t = pm 4]Wait, but the options include -4, -2, 0, and 4. So, (t) can be either 4 or -4. That would correspond to options A and D.But hold on, the problem mentions that (x) is in the set of all complex numbers. Does that affect the result? I think not, because even if the roots are complex, the formula for the difference still holds because it's derived from Vieta's formulas, which apply to any quadratic equation, whether the roots are real or complex.But let me double-check. If the roots are complex, they are conjugates, so their difference would be purely imaginary. However, the absolute value (or modulus) of a purely imaginary number (bi) is (|b|). So, in this case, (|x_1 - x_2| = 2sqrt{2}) would still be valid because it's the modulus of the difference.So, my initial conclusion stands: (t = pm 4), which are options A and D.But wait, the answer given earlier said A, C, D. Why is 0 included? Let me think again.Is there a possibility that (t = 0) also satisfies the condition? Let's test (t = 0).If (t = 0), the equation becomes (x^2 + 2 = 0), so the roots are (x = pm sqrt{-2} = pm isqrt{2}). The difference between the roots is (isqrt{2} - (-isqrt{2}) = 2isqrt{2}). The modulus of this difference is (|2isqrt{2}| = 2sqrt{2}), which matches the given condition.Oh, so (t = 0) also works. That means the possible values of (t) are -4, 0, and 4. So, options A, C, and D.But why did my initial calculation only give me (t = pm 4)? Because when I solved (t^2 = 16), I got (t = pm 4). But when (t = 0), it's a different case where the roots are purely imaginary, and the difference still satisfies the modulus condition.So, I need to consider both cases: when the roots are real and when they are complex.Wait, but in the case of real roots, the discriminant must be non-negative. The discriminant (D = t^2 - 8). For real roots, (D geq 0), so (t^2 geq 8), which means (t geq 2sqrt{2}) or (t leq -2sqrt{2}). Since (2sqrt{2}) is approximately 2.828, so (t) must be greater than or equal to about 2.828 or less than or equal to -2.828.But in the options, the closest values are -4, -2, 0, 4. So, -4 and 4 are within the range where the roots are real, and -2 and 0 are within the range where the roots are complex.But when (t = 0), as I saw earlier, the roots are purely imaginary, and the modulus of their difference is (2sqrt{2}), which satisfies the condition.Similarly, when (t = -2), let's check:If (t = -2), the equation becomes (x^2 - 2x + 2 = 0). The discriminant is ((-2)^2 - 4 times 1 times 2 = 4 - 8 = -4), so the roots are complex: (x = [2 pm sqrt{-4}]/2 = 1 pm i). The difference between the roots is (2i), and the modulus is (2), which is less than (2sqrt{2}). So, (t = -2) does not satisfy the condition.Similarly, if (t = 2), which is not an option, the roots would be (1 pm i), same as above, modulus of difference is 2.So, only (t = -4), (t = 0), and (t = 4) satisfy the condition (|x_1 - x_2| = 2sqrt{2}).Wait, but when (t = 4), the equation is (x^2 + 4x + 2 = 0). The discriminant is (16 - 8 = 8), so roots are ([-4 pm sqrt{8}]/2 = -2 pm sqrt{2}). The difference between the roots is (2sqrt{2}), which matches the condition.Similarly, when (t = -4), the equation is (x^2 - 4x + 2 = 0). The discriminant is (16 - 8 = 8), so roots are ([4 pm sqrt{8}]/2 = 2 pm sqrt{2}). The difference is again (2sqrt{2}).And when (t = 0), as before, the roots are (pm isqrt{2}), difference is (2isqrt{2}), modulus is (2sqrt{2}).So, indeed, (t = -4), (t = 0), and (t = 4) satisfy the condition. Therefore, the correct options are A, C, and D.I think initially, I only considered the real roots case, but the problem allows for complex roots as well, so I need to include the case where (t = 0), which gives purely imaginary roots with the required modulus difference.So, the possible values of (t) are -4, 0, and 4.

💡Okay, so I have two problems here about coordinate geometry. The first one is about finding the equation of a median in a triangle, and the second one is about finding the equation of a circle given some conditions. Let me try to tackle them one by one.Starting with the first problem: Triangle ABC has vertices at A(4,1), B(0,3), and C(2,4). I need to find the equation of the median BD, where D is the midpoint of side AC. Hmm, okay, so a median connects a vertex to the midpoint of the opposite side. So, in this case, BD connects vertex B to the midpoint D of side AC.First, I need to find the coordinates of point D. Since D is the midpoint of AC, I can use the midpoint formula. The midpoint formula is ((x1 + x2)/2, (y1 + y2)/2). So, plugging in the coordinates of A and C:x-coordinate of D: (4 + 2)/2 = 6/2 = 3y-coordinate of D: (1 + 4)/2 = 5/2 = 2.5So, D is at (3, 2.5). Got that.Now, I need to find the equation of the line BD. Point B is at (0,3), and point D is at (3, 2.5). To find the equation of a line, I can use the two-point form. The formula is (y - y1) = m(x - x1), where m is the slope.First, let me find the slope m between points B(0,3) and D(3, 2.5). The slope formula is (y2 - y1)/(x2 - x1).So, m = (2.5 - 3)/(3 - 0) = (-0.5)/3 = -1/6.Okay, so the slope is -1/6. Now, using point B(0,3) to write the equation:y - 3 = (-1/6)(x - 0)Simplifying, y = (-1/6)x + 3.But the problem asks for the equation in standard form. Standard form is Ax + By + C = 0. So, let me rearrange the equation:y = (-1/6)x + 3Multiply both sides by 6 to eliminate the fraction:6y = -x + 18Bring all terms to one side:x + 6y - 18 = 0So, the equation of the median BD in standard form is x + 6y - 18 = 0. That seems right. Let me double-check my calculations.Midpoint D: (3, 2.5) – correct. Slope between B(0,3) and D(3,2.5): (2.5 - 3)/(3 - 0) = (-0.5)/3 = -1/6 – correct. Equation using point B: y - 3 = (-1/6)(x - 0) → y = (-1/6)x + 3 – correct. Converting to standard form: x + 6y - 18 = 0 – yes, that looks good.Alright, moving on to the second problem. A circle C has its center at the intersection of the lines 2x + y + 1 = 0 and x + 3y - 4 = 0. It's also tangent to the line 3x + 4y + 17 = 0. I need to find the equation of circle C.First, I need to find the center of the circle, which is the intersection point of the two given lines. So, I have to solve the system of equations:2x + y + 1 = 0 ...(1)x + 3y - 4 = 0 ...(2)Let me solve equation (2) for x: x = 4 - 3y.Now, substitute this into equation (1):2(4 - 3y) + y + 1 = 08 - 6y + y + 1 = 09 - 5y = 0-5y = -9y = 9/5 = 1.8Now, substitute y = 9/5 back into x = 4 - 3y:x = 4 - 3*(9/5) = 4 - 27/5 = (20/5 - 27/5) = -7/5 = -1.4So, the center of the circle is at (-7/5, 9/5). Got that.Next, the circle is tangent to the line 3x + 4y + 17 = 0. The distance from the center to this line will be equal to the radius of the circle. So, I need to find the distance from point (-7/5, 9/5) to the line 3x + 4y + 17 = 0.The formula for the distance from a point (x0, y0) to the line Ax + By + C = 0 is |Ax0 + By0 + C| / sqrt(A^2 + B^2).Plugging in the values:A = 3, B = 4, C = 17x0 = -7/5, y0 = 9/5Distance = |3*(-7/5) + 4*(9/5) + 17| / sqrt(3^2 + 4^2)= |(-21/5) + (36/5) + 17| / 5= |(15/5) + 17| / 5= |3 + 17| / 5= |20| / 5= 4So, the radius is 4. Therefore, the equation of the circle is:(x - (-7/5))^2 + (y - 9/5)^2 = 4^2Simplify:(x + 7/5)^2 + (y - 9/5)^2 = 16Let me just double-check the calculations.Finding the center: Solved the system correctly. Substituted x = 4 - 3y into 2x + y +1=0, got y=9/5, then x=-7/5 – correct.Distance calculation: Plugged into the formula, computed numerator as |(-21/5 + 36/5 + 17)|. Let's compute step by step:3*(-7/5) = -21/54*(9/5) = 36/5So, -21/5 + 36/5 = 15/5 = 3Then, 3 + 17 = 20Absolute value is 20, divided by sqrt(9 + 16)=5, so 20/5=4 – correct.Therefore, the equation is (x + 7/5)^2 + (y - 9/5)^2 = 16. That looks good.I think both problems are solved correctly. The first one gave me the median equation in standard form, and the second one gave me the circle's equation with the center and radius calculated properly.**Final Answer**1. The equation of the median (BD) is (boxed{x + 6y - 18 = 0}).2. The equation of circle (C) is (boxed{left(x + frac{7}{5}right)^2 + left(y - frac{9}{5}right)^2 = 16}).

💡First, I need to determine the area of the ring-shaped lawn surrounding the circular television tower.The television tower has a base radius of 15 meters, and the lawn will be 5 meters wide around it. To find the area of the lawn, I'll calculate the area of the larger circle (including the tower and the lawn) and subtract the area of the smaller circle (just the tower).1. **Calculate the radius of the larger circle:** [ text{Radius of larger circle} = 15 text{ meters} + 5 text{ meters} = 20 text{ meters} ]2. **Calculate the area of the larger circle:** [ text{Area of larger circle} = pi times (20)^2 = 400pi text{ square meters} ]3. **Calculate the area of the smaller circle:** [ text{Area of smaller circle} = pi times (15)^2 = 225pi text{ square meters} ]4. **Determine the area of the lawn by subtracting the smaller area from the larger area:** [ text{Area of lawn} = 400pi - 225pi = 175pi text{ square meters} ]5. **Approximate the area using ( pi approx 3.14 ):** [ text{Area of lawn} approx 175 times 3.14 = 549.5 text{ square meters} ]Now, to find the total cost of planting the lawn:1. **Multiply the area of the lawn by the cost per square meter:** [ text{Total cost} = 549.5 text{ square meters} times 500 text{ yuan/square meter} = 274,750 text{ yuan} ]