💡First, I need to identify the smallest prime palindrome greater than 300. A palindrome reads the same forwards and backwards, so I'll start by listing palindromic numbers just above 300.The first few palindromes greater than 300 are 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, and so on. Since even numbers and numbers ending in 0, 2, 4, 5, 6, or 8 cannot be prime (except for the number 2, which is not relevant here), I can eliminate several candidates immediately.This leaves me with 313, 323, 333, 343, 353, 373, 383, 393, 404, etc. I can further eliminate numbers like 323, 333, 343, and 393 because they are divisible by 11.Now, I'm left with 313, 353, 373, 383, and 404. I'll check each of these to see if they are prime. - **313**: This number is a prime.- **353**: This number is also a prime.- **373**: This number is a prime.- **383**: This number is a prime.- **404**: This number is not a prime because it is even.Among these, the smallest prime palindrome greater than 300 is 313. Finally, to find the sum of the digits of 313, I add them together: 3 + 1 + 3 = 7.

💡First, I need to determine how many segments of 2/5 of a mile are in the total trip distance of 3.6 miles. To do this, I'll divide 3.6 by 2/5.Next, I'll calculate the cost associated with these segments by multiplying the number of segments by the charge per segment, which is 0.35.Finally, I'll subtract the total cost of the distance traveled from the overall trip charge to find the initial fee.

💡Okay, so I have this problem where I need to find the range of values for the expression (4left(x - frac{1}{2}right)^2 + (y - 1)^2 + 4xy) given that (x^2 + y^2 = 4). Hmm, that's a circle with radius 2 centered at the origin. So, x and y are points on this circle.First, I should probably try to simplify the given expression to make it easier to work with. Let me expand each term step by step.Starting with (4left(x - frac{1}{2}right)^2):[4left(x - frac{1}{2}right)^2 = 4left(x^2 - x + frac{1}{4}right) = 4x^2 - 4x + 1]Okay, that seems straightforward.Next, expanding ((y - 1)^2):[(y - 1)^2 = y^2 - 2y + 1]Alright, so now I have both expanded terms. Let me write them down together:[4x^2 - 4x + 1 + y^2 - 2y + 1]Combine like terms:[4x^2 + y^2 - 4x - 2y + 2]And then we have the last term, which is (4xy). Adding that in:[4x^2 + y^2 - 4x - 2y + 2 + 4xy]Hmm, so the entire expression simplifies to:[4x^2 + y^2 - 4x - 2y + 4xy + 2]I wonder if I can combine some of these terms or factor them in a way that makes it easier to analyze. Let me see.Looking at the quadratic terms: (4x^2 + y^2 + 4xy). That looks like a quadratic form. Maybe I can write it as a square or something similar. Let me try to complete the square or factor it.Let me group the quadratic terms:[4x^2 + 4xy + y^2]Wait, that's a perfect square:[(2x + y)^2]Yes! So, (4x^2 + 4xy + y^2 = (2x + y)^2). Nice, that simplifies things a bit.So, substituting back into the expression:[(2x + y)^2 - 4x - 2y + 2]Now, let's look at the linear terms: (-4x - 2y). Maybe I can factor something out here or relate it to the squared term.Let me see, (2x + y) is part of the squared term. If I can express (-4x - 2y) in terms of (2x + y), that might help.Indeed, (-4x - 2y = -2(2x + y)). Perfect! So now, the expression becomes:[(2x + y)^2 - 2(2x + y) + 2]Let me set (z = 2x + y) to make it even simpler. Then the expression is:[z^2 - 2z + 2]That's a quadratic in terms of z. Maybe I can complete the square here as well.Completing the square for (z^2 - 2z + 2):[(z^2 - 2z + 1) + 1 = (z - 1)^2 + 1]So, the entire expression simplifies to:[(z - 1)^2 + 1 = (2x + y - 1)^2 + 1]Wow, that's much simpler. So, the original expression is equal to ((2x + y - 1)^2 + 1). Now, since squares are always non-negative, the smallest value this expression can take is when ((2x + y - 1)^2) is zero, which would make the whole expression equal to 1. So, the minimum value is 1.But we need the range, so we also need the maximum value. To find the maximum, we need to find the maximum value of ((2x + y - 1)^2) given that (x^2 + y^2 = 4).Let me denote (k = 2x + y - 1). Then, the expression becomes (k^2 + 1). So, to find the range, I need to find the maximum and minimum values of (k) given the constraint (x^2 + y^2 = 4).So, (k = 2x + y - 1). Let's rearrange that:[2x + y = k + 1]This is a linear equation in x and y. The maximum and minimum values of (k + 1) (and hence (k)) occur when the line (2x + y = k + 1) is tangent to the circle (x^2 + y^2 = 4). The distance from the center of the circle (which is at (0,0)) to the line must be equal to the radius for tangency.The formula for the distance from a point ((x_0, y_0)) to the line (Ax + By + C = 0) is:[text{Distance} = frac{|Ax_0 + By_0 + C|}{sqrt{A^2 + B^2}}]In our case, the line is (2x + y - (k + 1) = 0), so (A = 2), (B = 1), and (C = -(k + 1)). The center of the circle is (0,0), so plugging into the distance formula:[frac{|2(0) + 1(0) - (k + 1)|}{sqrt{2^2 + 1^2}} = frac{| - (k + 1) |}{sqrt{5}} = frac{|k + 1|}{sqrt{5}}]For the line to be tangent to the circle, this distance must equal the radius, which is 2:[frac{|k + 1|}{sqrt{5}} = 2]Solving for (k + 1):[|k + 1| = 2sqrt{5}]So, (k + 1 = 2sqrt{5}) or (k + 1 = -2sqrt{5}). Therefore, (k = 2sqrt{5} - 1) or (k = -2sqrt{5} - 1).Thus, the maximum value of (k) is (2sqrt{5} - 1) and the minimum is (-2sqrt{5} - 1). Therefore, the maximum value of (k^2) is when (k) is at its maximum or minimum because squaring will give the same result for both positive and negative values, but since we're squaring, the maximum will be when (|k|) is maximum.Wait, actually, no. Let me think. Since (k) can be both positive and negative, but when we square it, the maximum of (k^2) occurs at the maximum absolute value of (k). So, the maximum of (k^2) is ((2sqrt{5} + 1)^2) or ((2sqrt{5} - 1)^2). Wait, actually, let's compute both.First, compute ((2sqrt{5} - 1)^2):[(2sqrt{5} - 1)^2 = (2sqrt{5})^2 - 2 times 2sqrt{5} times 1 + 1^2 = 20 - 4sqrt{5} + 1 = 21 - 4sqrt{5}]Next, compute ((-2sqrt{5} - 1)^2):[(-2sqrt{5} - 1)^2 = (2sqrt{5} + 1)^2 = (2sqrt{5})^2 + 2 times 2sqrt{5} times 1 + 1^2 = 20 + 4sqrt{5} + 1 = 21 + 4sqrt{5}]So, the maximum value of (k^2) is (21 + 4sqrt{5}), and the minimum is (21 - 4sqrt{5}). But wait, earlier I thought the minimum of the original expression was 1, which occurs when (k = 1). Hmm, that seems conflicting.Wait, no. Let me clarify. The expression is ((k)^2 + 1), so the minimum value occurs when (k = 0), giving 1, but actually, (k) can't be zero necessarily. Wait, no, because (k = 2x + y - 1), and since (x) and (y) are on the circle (x^2 + y^2 = 4), (k) can indeed be zero if (2x + y = 1) has a solution on the circle.But actually, from earlier, we found that (k) can range from (-2sqrt{5} - 1) to (2sqrt{5} - 1). So, the minimum value of (k) is (-2sqrt{5} - 1) and the maximum is (2sqrt{5} - 1). Therefore, the minimum value of (k^2) is when (k) is closest to zero, which would be when (k = 0) if possible, but since (k) can be negative, the minimum of (k^2) is actually the square of the minimum absolute value of (k). Wait, this is getting confusing.Let me step back. The expression is ((2x + y - 1)^2 + 1). The minimum value of this expression occurs when ((2x + y - 1)^2) is minimized, which is zero. So, the minimum value is 1. The maximum value occurs when ((2x + y - 1)^2) is maximized, which is when (|2x + y - 1|) is maximized.From earlier, we found that (2x + y) can range from (-2sqrt{5} - 1) to (2sqrt{5} - 1). Wait, no, actually, we found that (k = 2x + y - 1) ranges from (-2sqrt{5} - 1) to (2sqrt{5} - 1). Therefore, the maximum value of (|k|) is (2sqrt{5} + 1) because when (k = -2sqrt{5} - 1), the absolute value is (2sqrt{5} + 1), which is larger than (2sqrt{5} - 1).Therefore, the maximum value of (k^2) is ((2sqrt{5} + 1)^2 = 21 + 4sqrt{5}). So, the maximum value of the expression is (21 + 4sqrt{5} + 1 = 22 + 4sqrt{5}).Wait, no, hold on. The expression is ((k)^2 + 1), so if (k^2) is (21 + 4sqrt{5}), then the expression is (21 + 4sqrt{5} + 1 = 22 + 4sqrt{5}). Similarly, the minimum value is when (k^2) is zero, giving 1.Therefore, the range of the expression is from 1 to (22 + 4sqrt{5}).But let me double-check my steps to make sure I didn't make a mistake.1. Expanded the expression correctly.2. Recognized the quadratic form and completed the square.3. Set (z = 2x + y) and simplified to ((z - 1)^2 + 1).4. Expressed (k = z - 1 = 2x + y - 1).5. Found the range of (k) by using the distance from the center to the line, which gave (k) ranging from (-2sqrt{5} - 1) to (2sqrt{5} - 1).6. Then, computed (k^2) for both ends, realizing that the maximum (k^2) is when (k) is at its most negative, giving a larger square.7. Concluded that the expression ranges from 1 to (22 + 4sqrt{5}).Yes, that seems correct. So, the range is from 1 to (22 + 4sqrt{5}).

💡Okay, so I've got this math problem here: "One hundred forty-five more than the square of a number is the same as the square of the quantity that is 19 less than the number. What is the number?" Hmm, let me try to understand and solve this step by step.First, I need to translate this word problem into an algebraic equation. The problem mentions "a number," so I'll call that number ( x ). The phrase "one hundred forty-five more than the square of a number" translates to ( x^2 + 145 ). That makes sense because the square of the number is ( x^2 ), and adding 145 to it gives ( x^2 + 145 ).Next, the problem says "the square of the quantity that is 19 less than the number." The quantity that is 19 less than the number would be ( x - 19 ), and the square of that is ( (x - 19)^2 ).So, putting it all together, the equation becomes:[ x^2 + 145 = (x - 19)^2 ]Now, I need to solve this equation for ( x ). Let's start by expanding the right-hand side. Expanding ( (x - 19)^2 ) gives:[ (x - 19)^2 = x^2 - 38x + 361 ]I remember that ( (a - b)^2 = a^2 - 2ab + b^2 ), so applying that formula here, ( a = x ) and ( b = 19 ), so ( 2ab = 2 times x times 19 = 38x ), and ( b^2 = 19^2 = 361 ). So, that expansion seems correct.Now, substituting back into the equation:[ x^2 + 145 = x^2 - 38x + 361 ]Hmm, I notice that both sides have ( x^2 ). Maybe I can subtract ( x^2 ) from both sides to simplify the equation. Let's try that:[ x^2 + 145 - x^2 = x^2 - 38x + 361 - x^2 ]Simplifying both sides:[ 145 = -38x + 361 ]Okay, now I have a simpler equation:[ 145 = -38x + 361 ]I need to solve for ( x ). Let's get all the constants on one side and the term with ( x ) on the other. I'll subtract 361 from both sides:[ 145 - 361 = -38x ]Calculating ( 145 - 361 ):[ -216 = -38x ]Now, to solve for ( x ), I'll divide both sides by -38:[ frac{-216}{-38} = x ]Simplifying the negatives:[ frac{216}{38} = x ]Now, I can reduce this fraction. Let's see if 216 and 38 have a common divisor. 216 divided by 2 is 108, and 38 divided by 2 is 19. So, simplifying:[ frac{108}{19} = x ]So, ( x = frac{108}{19} ). Let me check if this makes sense. I'll plug this value back into the original equation to verify.First, calculate ( x^2 ):[ left( frac{108}{19} right)^2 = frac{11664}{361} ]Then, ( x^2 + 145 ):[ frac{11664}{361} + 145 = frac{11664}{361} + frac{145 times 361}{361} = frac{11664 + 52345}{361} = frac{64009}{361} ]Now, calculate ( (x - 19)^2 ):[ left( frac{108}{19} - 19 right)^2 = left( frac{108}{19} - frac{361}{19} right)^2 = left( frac{-253}{19} right)^2 = frac{64009}{361} ]Both sides are equal, so ( x = frac{108}{19} ) is indeed the correct solution. I think that's it!

💡Okay, so I have this series: 1 + 3 + 9 + 27 + 81 + ... + 19683. I need to find the sum of all these terms. Hmm, let me think about how to approach this.First, I remember that this looks like a geometric series. A geometric series is one where each term is multiplied by a constant ratio to get the next term. Let me check the ratio between the terms. Starting from 1, the next term is 3. So, 3 divided by 1 is 3. Then, 9 divided by 3 is also 3. Similarly, 27 divided by 9 is 3, and so on. So, the common ratio, which I'll call r, is 3. That makes sense.Now, the first term, which I'll call a, is 1. The last term given is 19683. I need to figure out how many terms are in this series. I remember that in a geometric series, the nth term can be found using the formula:a_n = a * r^(n-1)Here, a_n is the nth term, which is 19683. Plugging in the values I have:19683 = 1 * 3^(n-1)So, 3^(n-1) = 19683. Hmm, I need to solve for n. I think I can do this by expressing 19683 as a power of 3. Let me see:3^1 = 33^2 = 93^3 = 273^4 = 813^5 = 2433^6 = 7293^7 = 21873^8 = 65613^9 = 19683Oh, okay, so 3^9 = 19683. That means:3^(n-1) = 3^9Since the bases are the same, the exponents must be equal:n - 1 = 9So, n = 10. There are 10 terms in this series.Now, to find the sum of a geometric series, I remember the formula:S_n = a * (r^n - 1) / (r - 1)Where S_n is the sum of the first n terms. Plugging in the values I have:S_10 = 1 * (3^10 - 1) / (3 - 1)First, let me calculate 3^10. I know that 3^9 is 19683, so 3^10 is 3 * 19683, which is 59049.So, S_10 = (59049 - 1) / (2) = 59048 / 2 = 29524.Wait, let me double-check that. 59049 minus 1 is indeed 59048, and dividing that by 2 gives 29524. That seems right.Just to make sure I didn't make a mistake, let me recount the number of terms. Starting from 1 as the first term, each subsequent term is multiplied by 3. So, term 1: 1, term 2: 3, term 3: 9, term 4: 27, term 5: 81, term 6: 243, term 7: 729, term 8: 2187, term 9: 6561, term 10: 19683. Yep, that's 10 terms. So, n=10 is correct.And the common ratio r is 3, which I confirmed earlier. So, the formula should apply correctly.Another way to think about it is to add the terms step by step, but that would be tedious with 10 terms. However, for verification, let me try adding a few terms and see if it aligns with the formula.1 + 3 = 44 + 9 = 1313 + 27 = 4040 + 81 = 121121 + 243 = 364364 + 729 = 10931093 + 2187 = 32803280 + 6561 = 98419841 + 19683 = 29524Wow, that actually matches the result from the formula. So, that gives me more confidence that 29524 is the correct sum.I think I've covered all the steps and double-checked my work. It seems solid. So, the sum of the series is 29524.

💡First, determine each person's work rate based on their efficiency compared to Mary.Mary's work rate is ( frac{1}{28} ) of the work per day.Rosy is 40% more efficient than Mary, so her work rate is:[ frac{1}{28} times 1.40 = frac{1.40}{28} = frac{1}{20} ]Tim is 20% less efficient than Mary, so his work rate is:[ frac{1}{28} times 0.80 = frac{0.80}{28} = frac{1}{35} ]Next, add their work rates to find the combined work rate:[ frac{1}{28} + frac{1}{20} + frac{1}{35} ]Convert each fraction to have a common denominator of 140:[ frac{5}{140} + frac{7}{140} + frac{4}{140} = frac{16}{140} = frac{2}{35} ]Finally, calculate the total days needed by taking the reciprocal of the combined work rate:[ frac{1}{frac{2}{35}} = frac{35}{2} = 17.5 text{ days} ]