💡Okay, so I have this problem here: I need to find the largest possible difference between two different prime numbers that add up to 172. The options given are A) 150, B) 155, C) 162, D) 165, and E) 170. Hmm, interesting. Let me think about how to approach this.First, I remember that any even number greater than 2 can be expressed as the sum of two prime numbers. This is known as the Goldbach Conjecture, although it's still just a conjecture, it's been tested for very large numbers and seems to hold true. So, for 172, which is even and definitely larger than 2, there should be at least one pair of primes that add up to it.But the question isn't just asking for any pair; it's asking for the pair with the largest possible difference between them. That means I need to find two primes, let's call them p and q, such that p + q = 172, and the absolute difference |p - q| is as large as possible.To maximize the difference between p and q, I should aim to have one prime as small as possible and the other as large as possible. That way, their difference will be the biggest. So, I'll start by checking the smallest prime numbers and see if their complement (172 minus the small prime) is also a prime.Let's list out the small primes: 2, 3, 5, 7, 11, 13, 17, and so on. I'll subtract each of these from 172 and check if the result is a prime number.Starting with the smallest prime, 2:172 - 2 = 170. Is 170 a prime? Well, 170 is even and greater than 2, so it's divisible by 2. Therefore, 170 is not a prime.Next, 3:172 - 3 = 169. Is 169 a prime? Hmm, 169 is 13 squared, so it's not a prime either.Moving on to 5:172 - 5 = 167. Is 167 a prime? Let me think. 167 is not divisible by 2, 3, 5, 7, 11, or 13. Let me check:- 167 ÷ 2 = 83.5 → Not divisible.- 167 ÷ 3 ≈ 55.666 → Not divisible.- 167 ÷ 5 = 33.4 → Not divisible.- 167 ÷ 7 ≈ 23.857 → Not divisible.- 167 ÷ 11 ≈ 15.181 → Not divisible.- 167 ÷ 13 ≈ 12.846 → Not divisible.Since 167 isn't divisible by any primes up to its square root (which is around 12.9), it must be a prime number. So, 5 and 167 are both primes, and they add up to 172.The difference between them is 167 - 5 = 162. That's a pretty large difference. But let me check if there's a pair with an even larger difference.The next prime after 5 is 7:172 - 7 = 165. Is 165 a prime? 165 is divisible by 5 (since it ends with a 5), so 165 ÷ 5 = 33. Therefore, 165 is not a prime.Next, 11:172 - 11 = 161. Is 161 a prime? Let's check:- 161 ÷ 7 = 23 → Exactly, so 161 is 7 × 23, which means it's not a prime.Moving on to 13:172 - 13 = 159. Is 159 a prime? 159 ÷ 3 = 53, so it's divisible by 3, hence not a prime.Next, 17:172 - 17 = 155. Is 155 a prime? 155 ÷ 5 = 31, so it's divisible by 5, not a prime.Continuing to 19:172 - 19 = 153. 153 ÷ 3 = 51, so it's divisible by 3, not a prime.Next, 23:172 - 23 = 149. Is 149 a prime? Let's check:- 149 ÷ 2 = 74.5 → Not divisible.- 149 ÷ 3 ≈ 49.666 → Not divisible.- 149 ÷ 5 = 29.8 → Not divisible.- 149 ÷ 7 ≈ 21.285 → Not divisible.- 149 ÷ 11 ≈ 13.545 → Not divisible.- 149 ÷ 13 ≈ 11.461 → Not divisible.Since 149 isn't divisible by any primes up to its square root (around 12.2), it's a prime number. So, 23 and 149 are both primes, and they add up to 172. The difference here is 149 - 23 = 126, which is less than 162. So, 162 is still the larger difference.Let me check a few more primes to be thorough.29:172 - 29 = 143. Is 143 a prime? 143 ÷ 11 = 13, so it's not a prime.31:172 - 31 = 141. 141 ÷ 3 = 47, so it's not a prime.37:172 - 37 = 135. 135 ÷ 5 = 27, so not a prime.41:172 - 41 = 131. Is 131 a prime? Let's check:- 131 ÷ 2 = 65.5 → Not divisible.- 131 ÷ 3 ≈ 43.666 → Not divisible.- 131 ÷ 5 = 26.2 → Not divisible.- 131 ÷ 7 ≈ 18.714 → Not divisible.- 131 ÷ 11 ≈ 11.909 → Not divisible.Since 131 isn't divisible by any primes up to its square root (around 11.45), it's a prime. So, 41 and 131 add up to 172, with a difference of 131 - 41 = 90. That's even smaller than 126.Continuing, 43:172 - 43 = 129. 129 ÷ 3 = 43, so it's not a prime.47:172 - 47 = 125. 125 ÷ 5 = 25, not a prime.53:172 - 53 = 119. 119 ÷ 7 = 17, so not a prime.59:172 - 59 = 113. Is 113 a prime? Let's check:- 113 ÷ 2 = 56.5 → Not divisible.- 113 ÷ 3 ≈ 37.666 → Not divisible.- 113 ÷ 5 = 22.6 → Not divisible.- 113 ÷ 7 ≈ 16.142 → Not divisible.- 113 ÷ 11 ≈ 10.272 → Not divisible.So, 113 is a prime. Therefore, 59 and 113 add up to 172, with a difference of 113 - 59 = 54. Still smaller.I could keep going, but I can see a pattern here. As the smaller prime increases, the larger prime decreases, and the difference between them also decreases. So, the largest difference we found so far is 162 with the primes 5 and 167.Just to make sure I haven't missed any primes between 2 and 5, let's check 2 again:172 - 2 = 170, which is not a prime, as established earlier. So, 2 doesn't work.Therefore, the pair with the largest difference is 5 and 167, giving a difference of 162. Looking at the options, that's option C.I think that's the correct answer. It makes sense because starting from the smallest prime and moving upwards, the first valid pair we find gives the largest possible difference. Any larger primes would result in a smaller difference when paired with their complement.

💡Okay, so I have this inequality: ( x^2 - 4a x + 3a^2 < 0 ), and it's given that ( a < 0 ). The solution set is an interval ( (x_1, x_2) ). I need to find the maximum value of the expression ( x_1 + x_2 + frac{a}{x_1 x_2} ). Hmm, let me break this down step by step.First, I remember that for a quadratic inequality like ( ax^2 + bx + c < 0 ), the solution set is between the two roots if the quadratic opens upwards (which it does here since the coefficient of ( x^2 ) is positive). So, the solution set ( (x_1, x_2) ) means ( x_1 ) and ( x_2 ) are the roots of the equation ( x^2 - 4a x + 3a^2 = 0 ).Using Vieta's formulas, which relate the coefficients of a polynomial to sums and products of its roots, I can find ( x_1 + x_2 ) and ( x_1 x_2 ). For the quadratic equation ( x^2 - 4a x + 3a^2 = 0 ), the sum of the roots ( x_1 + x_2 ) is equal to ( frac{-b}{a} ). Here, ( a = 1 ) (the coefficient of ( x^2 )), and ( b = -4a ). So, ( x_1 + x_2 = frac{-(-4a)}{1} = 4a ).Similarly, the product of the roots ( x_1 x_2 ) is ( frac{c}{a} ). In this case, ( c = 3a^2 ), so ( x_1 x_2 = frac{3a^2}{1} = 3a^2 ).So, now I have ( x_1 + x_2 = 4a ) and ( x_1 x_2 = 3a^2 ). The expression I need to maximize is ( x_1 + x_2 + frac{a}{x_1 x_2} ). Substituting the values from Vieta's formulas, this becomes:( 4a + frac{a}{3a^2} ).Simplifying the second term: ( frac{a}{3a^2} = frac{1}{3a} ). So, the expression is:( 4a + frac{1}{3a} ).Now, I need to find the maximum value of this expression given that ( a < 0 ). Let me denote ( f(a) = 4a + frac{1}{3a} ). Since ( a ) is negative, both terms ( 4a ) and ( frac{1}{3a} ) are negative. So, ( f(a) ) is negative.To find the maximum value of ( f(a) ), which is the least negative value, I can use calculus. Alternatively, since it's a function of a single variable, I can find its critical points by taking the derivative and setting it equal to zero.Let's compute the derivative ( f'(a) ):( f'(a) = frac{d}{da} left( 4a + frac{1}{3a} right) = 4 - frac{1}{3a^2} ).Set the derivative equal to zero to find critical points:( 4 - frac{1}{3a^2} = 0 ).Solving for ( a ):( 4 = frac{1}{3a^2} )Multiply both sides by ( 3a^2 ):( 12a^2 = 1 )Divide both sides by 12:( a^2 = frac{1}{12} )Take the square root:( a = pm frac{1}{2sqrt{3}} )But since ( a < 0 ), we take the negative root:( a = -frac{1}{2sqrt{3}} )Now, let's check if this critical point is a maximum. Since ( f(a) ) is defined for ( a < 0 ), we can analyze the second derivative or test intervals around the critical point.Compute the second derivative ( f''(a) ):( f''(a) = frac{d}{da} left( 4 - frac{1}{3a^2} right) = 0 + frac{2}{3a^3} )At ( a = -frac{1}{2sqrt{3}} ), ( f''(a) = frac{2}{3(-frac{1}{2sqrt{3}})^3} ).Calculate the denominator:( (-frac{1}{2sqrt{3}})^3 = -frac{1}{8 cdot 3sqrt{3}} = -frac{1}{24sqrt{3}} )So, ( f''(a) = frac{2}{3 cdot (-frac{1}{24sqrt{3}})} = frac{2}{ -frac{1}{8sqrt{3}}} = -16sqrt{3} )Since ( f''(a) < 0 ), the function is concave down at this point, indicating a local maximum.Therefore, the maximum value of ( f(a) ) occurs at ( a = -frac{1}{2sqrt{3}} ).Now, let's compute ( f(a) ) at this value:( fleft(-frac{1}{2sqrt{3}}right) = 4 left(-frac{1}{2sqrt{3}}right) + frac{1}{3 left(-frac{1}{2sqrt{3}}right)} )Simplify each term:First term: ( 4 times -frac{1}{2sqrt{3}} = -frac{2}{sqrt{3}} )Second term: ( frac{1}{3 times -frac{1}{2sqrt{3}}} = frac{1}{ -frac{3}{2sqrt{3}}} = -frac{2sqrt{3}}{3} )So, adding both terms:( -frac{2}{sqrt{3}} - frac{2sqrt{3}}{3} )To combine these, let's rationalize the denominators:( -frac{2}{sqrt{3}} = -frac{2sqrt{3}}{3} )So, the expression becomes:( -frac{2sqrt{3}}{3} - frac{2sqrt{3}}{3} = -frac{4sqrt{3}}{3} )Therefore, the maximum value of ( x_1 + x_2 + frac{a}{x_1 x_2} ) is ( -frac{4sqrt{3}}{3} ).Wait, but looking back at the options, option D is ( -frac{4sqrt{3}}{3} ), which matches my result. So, I think that's the correct answer.Alternatively, I could have used the AM-GM inequality, but since the expression involves both ( a ) and ( 1/a ), and ( a ) is negative, it's a bit tricky. The calculus approach seems more straightforward here.Just to recap:1. Found the roots using Vieta's formulas.2. Expressed the target expression in terms of ( a ).3. Took the derivative to find critical points.4. Determined that the critical point is a maximum.5. Calculated the value at that point.6. Confirmed the result matches one of the options.I think that covers all the steps. I don't see any mistakes in my reasoning, so I'm confident that D is the correct answer.**Final Answer**boxed{D}

💡First, I need to identify the highest and lowest scores from the given data set. The scores are 90, 89, 90, 95, 93, 94, and 93. The highest score is 95, and the lowest score is 89.Next, I'll remove these extreme values to focus on the middle scores: 90, 90, 93, 94, and 93.To find the average, I'll sum these remaining scores and divide by the number of scores. The sum is 90 + 90 + 93 + 94 + 93, which equals 460. Dividing this by 5 gives an average of 92.For the variance, I'll calculate the squared difference between each score and the average, then find the average of these squared differences. The squared differences are (90-92)² = 4, (90-92)² = 4, (93-92)² = 1, (94-92)² = 4, and (93-92)² = 1. Summing these gives 14. Dividing by 5 results in a variance of 2.8.Therefore, the average is 92 and the variance is 2.8, which corresponds to option B.

💡First, I need to determine the total area of the grid. Since the grid is formed by squares of (1 , text{cm}) on each side and it appears to be a (5 times 5) grid, the total area is:[5 , text{cm} times 5 , text{cm} = 25 , text{cm}^2]Next, I'll calculate the area of the shaded regions. There are two types of shaded triangles:1. **Triangles with base (1 , text{cm}) and height (2 , text{cm}):** [ text{Area} = frac{1}{2} times 1 , text{cm} times 2 , text{cm} = 1 , text{cm}^2 ] There are two such triangles, so their combined area is: [ 2 times 1 , text{cm}^2 = 2 , text{cm}^2 ]2. **Triangles with base (1 , text{cm}) and height (3 , text{cm}):** [ text{Area} = frac{1}{2} times 1 , text{cm} times 3 , text{cm} = 1.5 , text{cm}^2 ] There are two such triangles, so their combined area is: [ 2 times 1.5 , text{cm}^2 = 3 , text{cm}^2 ]Adding these together gives the total shaded area:[2 , text{cm}^2 + 3 , text{cm}^2 = 5 , text{cm}^2]The unshaded area is the total area minus the shaded area:[25 , text{cm}^2 - 5 , text{cm}^2 = 20 , text{cm}^2]Finally, the ratio of the shaded area to the unshaded area is:[frac{5 , text{cm}^2}{20 , text{cm}^2} = frac{1}{4}]

💡First, I need to determine the original price of the computer, which is denoted as ( d ) dollars. I know that ( 2d = 580 ). To find ( d ), I can divide both sides of the equation by 2:[d = frac{580}{2} = 290]Now that I have the original price (( d = 290 ) dollars), I need to calculate the new price after a 30% increase. To find the amount of the increase, I multiply the original price by 30%:[text{Increase} = 290 times 0.30 = 87]Finally, I add the increase to the original price to find the new price:[text{New price} = 290 + 87 = 377]So, the new price of the computer after a 30% increase is (boxed{377}) dollars.

💡Okay, so I need to figure out the volume of a regular octahedron with edge length ( a ). Hmm, I remember that an octahedron is like two square pyramids stuck together at their bases. So, maybe I can find the volume of one pyramid and then double it to get the total volume of the octahedron.First, let me recall the formula for the volume of a pyramid. It's ( frac{1}{3} times text{base area} times text{height} ). So, I need to find the base area and the height of one of these pyramids.The base of each pyramid is a square, right? And since the edge length of the octahedron is ( a ), the base of the pyramid must also have an edge length of ( a ). So, the area of the base ( S ) is ( a times a = a^2 ).Now, I need to find the height ( h ) of the pyramid. This is a bit trickier. I think I can use some geometry here. If I consider the octahedron, it's symmetric, so the height of the pyramid should be related to the edge length ( a ).Let me visualize the octahedron. It has eight triangular faces, all equilateral triangles. Each face is an equilateral triangle with side length ( a ). The octahedron can be inscribed in a sphere, but I'm not sure if that helps directly.Wait, maybe I can think about the distance from the center of the octahedron to one of its vertices. That should be the height of the pyramid. How can I find that?I remember that in a regular octahedron, the distance from the center to a vertex is related to the edge length. Maybe I can use the Pythagorean theorem somewhere.Let me consider one of the pyramids. The base is a square with side length ( a ), and the apex is the top vertex of the octahedron. The height ( h ) of the pyramid is the distance from the apex to the center of the base.If I can find this height, I can plug it into the volume formula.To find ( h ), I can consider the right triangle formed by the height ( h ), half of the diagonal of the base square, and the edge length ( a ).First, let's find the diagonal of the base square. The diagonal ( d ) of a square with side length ( a ) is ( asqrt{2} ). So, half of the diagonal is ( frac{asqrt{2}}{2} ).Now, in the right triangle, one leg is ( h ), the other leg is ( frac{asqrt{2}}{2} ), and the hypotenuse is the edge length ( a ).So, by the Pythagorean theorem:[h^2 + left( frac{asqrt{2}}{2} right)^2 = a^2]Let me compute ( left( frac{asqrt{2}}{2} right)^2 ):[left( frac{asqrt{2}}{2} right)^2 = frac{2a^2}{4} = frac{a^2}{2}]So, plugging back into the equation:[h^2 + frac{a^2}{2} = a^2]Subtract ( frac{a^2}{2} ) from both sides:[h^2 = a^2 - frac{a^2}{2} = frac{a^2}{2}]Therefore, ( h = sqrt{frac{a^2}{2}} = frac{a}{sqrt{2}} ).Okay, so the height of the pyramid is ( frac{a}{sqrt{2}} ).Now, let's compute the volume of one pyramid:[V_{text{pyramid}} = frac{1}{3} times text{base area} times text{height} = frac{1}{3} times a^2 times frac{a}{sqrt{2}} = frac{a^3}{3sqrt{2}}]But since the octahedron is made up of two such pyramids, the total volume ( V ) is:[V = 2 times V_{text{pyramid}} = 2 times frac{a^3}{3sqrt{2}} = frac{2a^3}{3sqrt{2}}]Hmm, this expression can be rationalized. Let me multiply numerator and denominator by ( sqrt{2} ):[V = frac{2a^3 times sqrt{2}}{3 times 2} = frac{a^3 sqrt{2}}{3}]So, the volume of the regular octahedron with edge length ( a ) is ( frac{a^3 sqrt{2}}{3} ).Wait, let me double-check my steps to make sure I didn't make a mistake. I found the base area correctly as ( a^2 ). Then, I used the Pythagorean theorem to find the height of the pyramid, which seems right. I got ( h = frac{a}{sqrt{2}} ), which makes sense because in a regular octahedron, the height should be related to the edge length by that factor.Then, calculating the volume of the pyramid, I multiplied ( frac{1}{3} times a^2 times frac{a}{sqrt{2}} ), which gives ( frac{a^3}{3sqrt{2}} ). Doubling that for the two pyramids gives ( frac{2a^3}{3sqrt{2}} ), and rationalizing the denominator gives ( frac{a^3 sqrt{2}}{3} ).Yes, that seems correct. I think I did it right.