💡Okay, so I have this problem about a rectangular parallelepiped, which I think is just a fancy name for a rectangular box. The problem says that the lengths of segments EH, HG, and EG are consecutive integers, and the height of the parallelepiped is 12. I need to find the volume of this box.First, I should probably visualize the figure. Since it's a rectangular box, I imagine it has edges of different lengths. The segments EH, HG, and EG are probably edges or face diagonals of the box. The figure is referenced, but since I don't have the image, I'll have to work it out mentally.Let me recall that in a rectangular box, the space diagonal can be found using the Pythagorean theorem in three dimensions. But here, the problem mentions segments EH, HG, and EG. These could be edges or face diagonals. Since they are consecutive integers, their lengths are n, n+1, n+2 for some integer n.Wait, the problem says EH, HG, and EG are consecutive integers. So, perhaps EH and HG are edges, and EG is a face diagonal? Or maybe EH and HG are edges, and EG is the space diagonal? Hmm, I need to figure out which one it is.If I consider EH and HG as edges meeting at point H, then EG would be the diagonal connecting E and G. Depending on where E and G are, EG could be a face diagonal or a space diagonal. If E and G are on the same face, then EG is a face diagonal. If they are on opposite corners of the box, then EG is the space diagonal.But the problem says they are consecutive integers. Let's assume that EH and HG are edges, and EG is the face diagonal. Then, by the Pythagorean theorem, EH² + HG² = EG². Since they are consecutive integers, let's denote EH = a, HG = a+1, and EG = a+2. Then:a² + (a+1)² = (a+2)²Let me compute that:a² + (a² + 2a + 1) = a² + 4a + 4Simplify:2a² + 2a + 1 = a² + 4a + 4Subtract a² + 4a + 4 from both sides:2a² + 2a + 1 - a² - 4a - 4 = 0Which simplifies to:a² - 2a - 3 = 0Now, solving this quadratic equation:a = [2 ± sqrt(4 + 12)] / 2 = [2 ± sqrt(16)] / 2 = [2 ± 4]/2So, a = (2 + 4)/2 = 3 or a = (2 - 4)/2 = -1Since length can't be negative, a = 3.So, EH = 3, HG = 4, and EG = 5. That makes sense because 3, 4, 5 is a Pythagorean triple.Wait, so if EH and HG are 3 and 4, then the edges of the base are 3 and 4, and the height is given as 12. So, the volume would be length × width × height = 3 × 4 × 12.Calculating that: 3 × 4 = 12, and 12 × 12 = 144.So, the volume is 144.But hold on, let me make sure I didn't make any wrong assumptions. I assumed that EH and HG are edges, and EG is the face diagonal. But what if EG is the space diagonal instead?If EG is the space diagonal, then the formula would be EH² + HG² + height² = EG². But in this case, the height is given as 12, which is different from the other segments. So, if EH, HG, and EG are consecutive integers, and the height is 12, then maybe EH and HG are edges, and EG is the space diagonal.Let me test that scenario.Let EH = a, HG = a+1, EG = a+2, and height = 12.Then, the space diagonal EG would satisfy:EH² + HG² + height² = EG²So, a² + (a+1)² + 12² = (a+2)²Compute that:a² + (a² + 2a + 1) + 144 = a² + 4a + 4Simplify:2a² + 2a + 1 + 144 = a² + 4a + 4Which is:2a² + 2a + 145 = a² + 4a + 4Subtract a² + 4a + 4 from both sides:2a² + 2a + 145 - a² - 4a - 4 = 0Simplify:a² - 2a + 141 = 0Now, solving this quadratic:a = [2 ± sqrt(4 - 564)] / 2But sqrt(4 - 564) is sqrt(-560), which is imaginary. So, that can't be.Therefore, my initial assumption must be correct: EH and HG are edges, and EG is the face diagonal. So, the edges are 3, 4, and the height is 12. So, volume is 3×4×12=144.Wait, but just to be thorough, could EG be another diagonal? Maybe not the face diagonal but a different one? Hmm, in a rectangular box, the face diagonals are on each face, and the space diagonal goes from one corner to the opposite corner through the interior.Given that EH and HG are edges, and EG is a diagonal, it's more likely a face diagonal because the space diagonal would involve the height as well, which complicates things, and we saw that leads to an imaginary solution.So, yeah, I think my first approach is correct. The volume is 144.**Final Answer**The volume of the parallelepiped is boxed{144}.

💡Okay, so I've got this problem about a triangle with sides a, b, and c. The relationship given is:frac{1}{a} - frac{1}{b} + frac{1}{c} = frac{1}{a - b + c}And I need to figure out what kind of triangle it is. The options are equilateral, isosceles with a as the base, isosceles with c as the base, scalene, or none of the above.Hmm, let's start by trying to simplify the equation. Maybe if I can manipulate it algebraically, I can find a relationship between a, b, and c that tells me about the triangle's type.First, let's write down the equation again:frac{1}{a} - frac{1}{b} + frac{1}{c} = frac{1}{a - b + c}I think I should find a common denominator for the left side to combine the fractions. The denominators are a, b, and c, so the common denominator would be abc. Let me rewrite each term with this common denominator.So,frac{1}{a} = frac{bc}{abc}, quad -frac{1}{b} = -frac{ac}{abc}, quad frac{1}{c} = frac{ab}{abc}Adding these together:frac{bc}{abc} - frac{ac}{abc} + frac{ab}{abc} = frac{bc - ac + ab}{abc}So the left side simplifies to:frac{bc - ac + ab}{abc}And that's equal to the right side:frac{1}{a - b + c}So now we have:frac{bc - ac + ab}{abc} = frac{1}{a - b + c}Hmm, maybe cross-multiplying will help. Let's do that:(bc - ac + ab)(a - b + c) = abcOkay, that looks a bit complicated, but let's expand the left side step by step.First, let me denote:Let’s call the first part (bc - ac + ab) as Term1 and the second part (a - b + c) as Term2.So, Term1 * Term2 = abc.Let me expand Term1 * Term2:Term1 = bc - ac + abTerm2 = a - b + cMultiplying each term in Term1 by each term in Term2:First, bc * a = abcbc * (-b) = -b^2cbc * c = bc^2Next, -ac * a = -a^2c-ac * (-b) = ab c-ac * c = -a c^2Then, ab * a = a^2bab * (-b) = -a b^2ab * c = a b cSo, putting all these together:abc - b^2c + bc^2 - a^2c + abc - a c^2 + a^2b - a b^2 + a b cNow, let's combine like terms:Looking for abc terms: There are three abc terms: abc, abc, abc. So that's 3abc.Next, terms with a^2b: Only one, which is +a^2b.Terms with a^2c: Only one, which is -a^2c.Terms with ab^2: Only one, which is -a b^2.Terms with b^2c: Only one, which is -b^2c.Terms with bc^2: Only one, which is +bc^2.Terms with a c^2: Only one, which is -a c^2.So, putting it all together:3abc + a^2b - a^2c - a b^2 - b^2c + bc^2 - a c^2Hmm, that's still quite a bit. Let me see if I can factor anything out or simplify further.Wait, maybe I made a mistake in expanding. Let me double-check.Wait, when I expanded Term1 * Term2, I think I might have missed some signs or terms. Let me go through it again carefully.Term1 = bc - ac + abTerm2 = a - b + cMultiply each term:First, bc * a = abcbc * (-b) = -b^2cbc * c = bc^2Next, -ac * a = -a^2c-ac * (-b) = +ab c-ac * c = -a c^2Then, ab * a = a^2bab * (-b) = -a b^2ab * c = a b cSo, that gives:abc - b^2c + bc^2 - a^2c + abc - a c^2 + a^2b - a b^2 + abcNow, combining like terms:abc appears three times: 3abca^2b: +a^2b-a^2c: -a^2c-ab^2: -a b^2-b^2c: -b^2cbc^2: +bc^2-a c^2: -a c^2So, yes, that seems correct.Now, let's write the equation:3abc + a^2b - a^2c - a b^2 - b^2c + bc^2 - a c^2 = abcSubtract abc from both sides:2abc + a^2b - a^2c - a b^2 - b^2c + bc^2 - a c^2 = 0Hmm, this is still quite complex. Maybe I can factor some terms.Let me group terms with a^2, b^2, c^2, etc.Looking at a^2 terms: a^2b - a^2c = a^2(b - c)Similarly, b^2 terms: -a b^2 - b^2c = -b^2(a + c)c^2 terms: bc^2 - a c^2 = c^2(b - a)And then we have 2abc.So, putting it all together:a^2(b - c) - b^2(a + c) + c^2(b - a) + 2abc = 0Hmm, maybe factor further.Let me see if I can factor out (b - c) or something similar.Wait, let's look at the terms:a^2(b - c) + c^2(b - a) - b^2(a + c) + 2abcHmm, maybe rearrange terms:a^2(b - c) + c^2(b - a) + 2abc - b^2(a + c)Let me see if I can factor (b - c) or (a - c) somewhere.Alternatively, maybe factor by grouping.Let me try grouping the first two terms and the last two terms:[a^2(b - c) + c^2(b - a)] + [2abc - b^2(a + c)]Looking at the first group:a^2(b - c) + c^2(b - a) = a^2b - a^2c + b c^2 - a c^2Hmm, not sure if that helps.Wait, maybe factor out (b - c) from the first two terms:a^2(b - c) + c^2(b - a) = (b - c)(a^2) + (b - a)(c^2)Hmm, not sure.Wait, maybe factor (b - a) from the second group:2abc - b^2(a + c) = b(2ac - b(a + c)) = b(2ac - a b - b c) = b(a(2c - b) - b c)Not sure.Alternatively, maybe factor (a - c):Looking at the equation:a^2(b - c) - b^2(a + c) + c^2(b - a) + 2abc = 0Let me rearrange:a^2(b - c) - b^2(a + c) + c^2(b - a) + 2abcHmm, perhaps factor (a - c):Wait, let me see:a^2(b - c) - b^2(a + c) + c^2(b - a) + 2abc= a^2b - a^2c - a b^2 - b^3 + b c^2 - a c^2 + 2abcHmm, maybe group terms differently.Let me group terms with a^2, b^2, c^2, and the rest:a^2b - a^2c - a b^2 - b^3 + b c^2 - a c^2 + 2abc= a^2(b - c) - b^2(a + b) + c^2(b - a) + 2abcWait, that doesn't seem helpful.Alternatively, maybe factor (a - b):Looking at the equation:a^2(b - c) - b^2(a + c) + c^2(b - a) + 2abcLet me see if (a - b) is a factor.Alternatively, maybe factor (a - c):Wait, let me try to factor (a - c):Looking at the equation:a^2(b - c) - b^2(a + c) + c^2(b - a) + 2abcLet me see:= a^2b - a^2c - a b^2 - b^3 + b c^2 - a c^2 + 2abcHmm, maybe factor (a - c) from some terms:a^2b - a^2c = a^2(b - c)Similarly, -a c^2 + b c^2 = c^2(b - a) = -c^2(a - b)Hmm, not sure.Wait, perhaps factor (a - c) from the entire equation.But I don't see an obvious way.Alternatively, maybe let's consider that the equation must hold for the sides of a triangle, so a, b, c must satisfy triangle inequalities.But maybe instead of expanding, I can approach the original equation differently.Let me go back to the original equation:frac{1}{a} - frac{1}{b} + frac{1}{c} = frac{1}{a - b + c}Let me denote S = a - b + c. Then the right side is 1/S.So, the equation becomes:frac{1}{a} - frac{1}{b} + frac{1}{c} = frac{1}{S}But S = a - b + c, so S = (a + c) - b.Hmm, not sure if that helps.Alternatively, maybe let me consider that in a triangle, the sum of any two sides must be greater than the third. So, a + c > b, which is true since S = a - b + c = (a + c) - b > 0.So, S is positive.Maybe I can write the equation as:frac{1}{a} + frac{1}{c} - frac{1}{b} = frac{1}{S}Where S = a - b + c.Hmm, maybe I can write S in terms of the other variables.Alternatively, maybe let me consider that if the triangle is isosceles, say a = c, then perhaps the equation simplifies.Let me test that.Assume a = c.Then, the equation becomes:frac{1}{a} - frac{1}{b} + frac{1}{a} = frac{1}{a - b + a}Simplify left side:frac{2}{a} - frac{1}{b} = frac{1}{2a - b}Hmm, let's see if this holds.Let me denote x = a, y = b.So,frac{2}{x} - frac{1}{y} = frac{1}{2x - y}Let me find a relationship between x and y.Cross-multiplying:left( frac{2}{x} - frac{1}{y} right) (2x - y) = 1Expand:frac{2(2x - y)}{x} - frac{2x - y}{y} = 1Simplify each term:First term: 2(2x - y)/x = (4x - 2y)/x = 4 - 2y/xSecond term: -(2x - y)/y = (-2x + y)/y = -2x/y + 1So, combining:4 - 2y/x - 2x/y + 1 = 1Simplify:(4 + 1) - 2y/x - 2x/y = 15 - 2y/x - 2x/y = 1Subtract 1:4 - 2y/x - 2x/y = 0Divide both sides by 2:2 - y/x - x/y = 0Multiply both sides by xy to eliminate denominators:2xy - y^2 - x^2 = 0Rearrange:-x^2 + 2xy - y^2 = 0Multiply both sides by -1:x^2 - 2xy + y^2 = 0Which factors as:(x - y)^2 = 0So, x = yWhich means a = bBut we assumed a = c, so a = b = cSo, the triangle would be equilateral.Wait, but the options don't include equilateral as the answer? Wait, no, option A is equilateral.But wait, in this case, assuming a = c led us to a = b = c, which is equilateral.But let me check if that's the only possibility.Alternatively, maybe the triangle is isosceles with c as the base, meaning a = b.Let me test that.Assume a = b.Then, the original equation becomes:frac{1}{a} - frac{1}{a} + frac{1}{c} = frac{1}{a - a + c}Simplify:0 + 1/c = 1/cWhich is true.So, if a = b, the equation holds.So, that suggests that the triangle could be isosceles with a = b, meaning c is the base.Wait, but earlier, assuming a = c led to a = b = c, which is a special case of isosceles.So, maybe the triangle is isosceles with a = b, which would make c the base.Alternatively, if a = c, then it's also isosceles, but with a = c, which would make b the base.Wait, but in the first case, assuming a = c led to a = b = c, which is equilateral, but in the second case, assuming a = b, the equation holds without any further constraints.So, perhaps the triangle must be isosceles with a = b, making c the base.Alternatively, maybe it's isosceles with c as the base, meaning a = b.Wait, let me think again.When I assumed a = c, I ended up with a = b = c, which is equilateral, but that's a specific case.When I assumed a = b, the equation holds for any c, as long as a = b.So, that suggests that the triangle must be isosceles with a = b, making c the base.Therefore, the answer would be option C: An isosceles triangle with c as the base.But wait, let me make sure.Alternatively, maybe the triangle is isosceles with a = c, but that led to a = b = c, which is equilateral, but that's a specific case.But in the case where a = b, the equation holds without any further constraints, so that's a more general case.Therefore, the triangle must be isosceles with a = b, making c the base.So, the answer is C.

💡Okay, so I have this problem here: Let the reciprocals of the roots of the quadratic equation (6x^2 - 11x + 7) be (gamma) and (delta). I need to evaluate (gamma + delta). Hmm, alright, let me think about how to approach this.First, I remember that for any quadratic equation of the form (ax^2 + bx + c = 0), there are some relationships between the coefficients and the roots. These are known as Vieta's formulas. I think Vieta's formulas state that if the roots are (r) and (s), then the sum of the roots (r + s) is equal to (-frac{b}{a}) and the product of the roots (rs) is equal to (frac{c}{a}). Let me verify that.Yes, that's correct. So, for the given equation (6x^2 - 11x + 7 = 0), the coefficients are (a = 6), (b = -11), and (c = 7). Therefore, the sum of the roots (r + s) should be (-frac{-11}{6}), which simplifies to (frac{11}{6}). And the product of the roots (rs) should be (frac{7}{6}). Okay, that makes sense.Now, the problem is asking about the reciprocals of the roots, which are (gamma = frac{1}{r}) and (delta = frac{1}{s}). I need to find (gamma + delta). Let me write that out:[gamma + delta = frac{1}{r} + frac{1}{s}]Hmm, to combine these fractions, I need a common denominator. The common denominator would be (rs), so:[gamma + delta = frac{s}{rs} + frac{r}{rs} = frac{r + s}{rs}]Oh, that's neat! So, (gamma + delta) is equal to (frac{r + s}{rs}). I already have expressions for (r + s) and (rs) from Vieta's formulas. Let me plug those values in.From earlier, I found that (r + s = frac{11}{6}) and (rs = frac{7}{6}). So substituting these into the equation:[gamma + delta = frac{frac{11}{6}}{frac{7}{6}}]Now, dividing these two fractions. Remember, dividing by a fraction is the same as multiplying by its reciprocal. So:[frac{frac{11}{6}}{frac{7}{6}} = frac{11}{6} times frac{6}{7}]I notice that the 6 in the numerator and the 6 in the denominator cancel each other out:[frac{11}{6} times frac{6}{7} = frac{11}{7}]So, (gamma + delta = frac{11}{7}). That seems straightforward. Let me just double-check my steps to make sure I didn't make any mistakes.1. Identified the quadratic equation and its coefficients.2. Applied Vieta's formulas to find the sum and product of the roots.3. Expressed the reciprocals of the roots and combined them into a single fraction.4. Substituted the known values from Vieta's formulas.5. Simplified the resulting expression.Everything seems to check out. I don't see any errors in my calculations. So, I'm confident that the value of (gamma + delta) is indeed (frac{11}{7}).**Final Answer**The value of (gamma + delta) is boxed{dfrac{11}{7}}.

💡Okay, so I need to find the sum of these two matrices. Hmm, let me make sure I remember how matrix addition works. I think it's pretty straightforward— you just add the corresponding elements from each matrix, right? So, if I have two matrices of the same size, I can add them element by element.Alright, let me look at the matrices given. The first matrix is:[begin{pmatrix} 4 & -3 0 & 5 end{pmatrix}]And the second matrix is:[begin{pmatrix} -6 & 8 7 & -10 end{pmatrix}]Both of these are 2x2 matrices, so they have the same dimensions, which means they can be added together. That's good; if they were different sizes, I wouldn't be able to add them.Now, let me recall how to add them. For each position in the matrix, I add the element from the first matrix to the element in the same position in the second matrix. So, for the element in the first row and first column, I add 4 and -6. For the first row and second column, I add -3 and 8, and so on for each element.Let me write this out step by step to make sure I don't make any mistakes.Starting with the first row, first column:4 (from the first matrix) + (-6) (from the second matrix) = 4 - 6 = -2Okay, that seems right. Now, first row, second column:-3 (from the first matrix) + 8 (from the second matrix) = -3 + 8 = 5Hmm, that looks correct. Moving on to the second row, first column:0 (from the first matrix) + 7 (from the second matrix) = 0 + 7 = 7And finally, the second row, second column:5 (from the first matrix) + (-10) (from the second matrix) = 5 - 10 = -5So, putting all these together, the resulting matrix after addition should be:[begin{pmatrix} -2 & 5 7 & -5 end{pmatrix}]Let me double-check my calculations to make sure I didn't make any errors. It's easy to mix up signs or add incorrectly, especially with negative numbers.First element: 4 + (-6) = -2. Yes, that's correct.Second element: -3 + 8 = 5. That's right.Third element: 0 + 7 = 7. Correct.Fourth element: 5 + (-10) = -5. Yes, that's accurate.So, all the additions seem to be correct. Therefore, the sum of the two matrices is indeed:[begin{pmatrix} -2 & 5 7 & -5 end{pmatrix}]I think that's it. Matrix addition isn't too complicated once you remember to add each corresponding element. It's just important to keep track of the signs to avoid mistakes. I feel confident that this is the correct answer.

💡First, I need to determine Aaron's current age. Let's denote Aaron's current age as ( A ) years.Three years ago, Mark was ( 28 - 3 = 25 ) years old. At that time, Mark's age was some years more than thrice Aaron's age. So, three years ago, Aaron's age was ( A - 3 ) years. The equation from the first statement is:[25 = 3(A - 3) + X]Where ( X ) is the number of years more than thrice Aaron's age that Mark was.Four years from now, Mark will be ( 28 + 4 = 32 ) years old. At that time, Mark's age will be 2 years more than twice Aaron's age. So, in four years, Aaron's age will be ( A + 4 ) years. The equation from the second statement is:[32 = 2(A + 4) + 2]Now I have two equations:1) ( 25 = 3(A - 3) + X )2) ( 32 = 2(A + 4) + 2 )Let's solve the second equation for ( A ):[32 = 2A + 8 + 2][32 = 2A + 10][2A = 32 - 10][2A = 22][A = 11]Now that I know Aaron's current age is 11 years, I can find ( X ) by plugging ( A ) into the first equation:[25 = 3(11 - 3) + X][25 = 3(8) + X][25 = 24 + X][X = 25 - 24][X = 1]So, three years ago, Mark's age was 1 year more than thrice Aaron's age.

💡Okay, so I have this problem where I need to find the intersection of two lines given by the equations 3y = -2x + 6 and -2y = 6x + 4. Hmm, intersection of two lines means the point where both lines meet, right? So, essentially, I need to find the values of x and y that satisfy both equations simultaneously.Alright, let me start by writing down the equations again to make sure I have them correctly:1. 3y = -2x + 62. -2y = 6x + 4I think the best way to approach this is by solving the system of equations. There are a couple of methods to solve a system of equations: substitution and elimination. I'm more comfortable with substitution, so I'll try that first.Looking at the first equation, 3y = -2x + 6, maybe I can solve for y in terms of x. That should make substitution easier. Let me do that.Dividing both sides by 3 to isolate y:y = (-2/3)x + 2Okay, so y is expressed in terms of x now. That means I can substitute this expression for y into the second equation. Let's write down the second equation again:-2y = 6x + 4Substituting y = (-2/3)x + 2 into this equation:-2[(-2/3)x + 2] = 6x + 4Now, I need to simplify this equation step by step. Let's start by distributing the -2 across the terms inside the brackets.First, multiply -2 by (-2/3)x:-2 * (-2/3)x = (4/3)xThen, multiply -2 by 2:-2 * 2 = -4So, putting it all together:(4/3)x - 4 = 6x + 4Now, I have an equation with x terms and constants. I need to solve for x. Let me get all the x terms on one side and the constants on the other.First, I'll add 4 to both sides to get rid of the -4 on the left:(4/3)x - 4 + 4 = 6x + 4 + 4Simplifying both sides:(4/3)x = 6x + 8Now, I need to get all the x terms on one side. Let me subtract 6x from both sides:(4/3)x - 6x = 8To combine these x terms, I need a common denominator. 6x can be written as (18/3)x, so:(4/3)x - (18/3)x = 8Subtracting the fractions:(-14/3)x = 8Now, I need to solve for x. To isolate x, I'll multiply both sides by the reciprocal of (-14/3), which is (-3/14):x = 8 * (-3/14)Multiplying these together:x = -24/14I can simplify this fraction by dividing numerator and denominator by 2:x = -12/7Okay, so x is -12/7. Now, I need to find the corresponding y value. I'll use the expression I found earlier for y in terms of x:y = (-2/3)x + 2Substituting x = -12/7 into this equation:y = (-2/3)*(-12/7) + 2First, multiply (-2/3) and (-12/7):(-2/3)*(-12/7) = (24/21)Simplify 24/21 by dividing numerator and denominator by 3:24/21 = 8/7So, y = 8/7 + 2Now, I need to add 8/7 and 2. To do this, I'll express 2 as a fraction with denominator 7:2 = 14/7So, y = 8/7 + 14/7 = 22/7Therefore, the y-coordinate is 22/7.Putting it all together, the intersection point is (-12/7, 22/7).Wait, let me double-check my calculations to make sure I didn't make any mistakes.Starting with the substitution:From the first equation: y = (-2/3)x + 2Substituted into the second equation:-2[(-2/3)x + 2] = 6x + 4Which simplifies to:(4/3)x - 4 = 6x + 4Then, adding 4 to both sides:(4/3)x = 6x + 8Subtracting 6x:(4/3)x - 6x = 8Convert 6x to thirds: 6x = 18/3 xSo, (4/3 - 18/3)x = 8Which is (-14/3)x = 8Multiply both sides by (-3/14):x = 8*(-3/14) = -24/14 = -12/7That seems correct.Then, plugging back into y:y = (-2/3)*(-12/7) + 2 = (24/21) + 2 = 8/7 + 14/7 = 22/7Yes, that checks out.So, the intersection point is indeed (-12/7, 22/7).I think that's the correct answer. I don't see any mistakes in my steps.