answer：First, I need to determine the total distance Julia traveled. She rode 45 miles at 15 miles per hour and 15 miles at 45 miles per hour. Adding these distances together gives a total of 60 miles.Next, I'll calculate the time taken for each segment of the trip. For the first segment, dividing the distance by the speed gives 45 miles divided by 15 miles per hour, which equals 3 hours. For the second segment, dividing 15 miles by 45 miles per hour results in 1/3 of an hour.Adding the times from both segments provides the total time for the trip: 3 hours plus 1/3 hour equals 10/3 hours.Finally, to find the average speed, I'll divide the total distance by the total time. Dividing 60 miles by 10/3 hours gives an average speed of 18 miles per hour.

answer：Okay, so I have this problem here: "Suppose that ( y^2 ) varies inversely with ( sqrt{z} ). If ( y = 3 ) when ( z = 16 ), find the value of ( z ) when ( y = 6 ). Express your answer in simplest fractional form."Hmm, let me try to understand what this means. The problem is talking about inverse variation. I remember that inverse variation means that as one quantity increases, the other decreases proportionally. The general form of an inverse variation is ( y = frac{k}{x} ), where ( k ) is a constant.But in this problem, it's not just ( y ) that's varying inversely with ( sqrt{z} ); it's ( y^2 ) that's varying inversely with ( sqrt{z} ). So, I think I can write this relationship as:[y^2 = frac{k}{sqrt{z}}]Where ( k ) is the constant of variation. That makes sense because if ( y^2 ) varies inversely with ( sqrt{z} ), then ( y^2 ) times ( sqrt{z} ) should be a constant. So, another way to write this is:[y^2 cdot sqrt{z} = k]Alright, so now I have this equation. The problem gives me a specific case where ( y = 3 ) and ( z = 16 ). I can use these values to find the constant ( k ).Let me plug those values into the equation:[(3)^2 cdot sqrt{16} = k]Calculating each part step by step. First, ( 3^2 ) is 9. Then, ( sqrt{16} ) is 4. So, multiplying those together:[9 cdot 4 = 36]So, ( k = 36 ). Got that down.Now, the problem asks for the value of ( z ) when ( y = 6 ). So, I need to use the same relationship but with ( y = 6 ) and ( k = 36 ) to find the new ( z ).Let me set up the equation again:[(6)^2 cdot sqrt{z} = 36]Calculating ( 6^2 ) gives me 36. So, plugging that in:[36 cdot sqrt{z} = 36]Hmm, okay, so I have ( 36 cdot sqrt{z} = 36 ). To solve for ( sqrt{z} ), I can divide both sides of the equation by 36:[sqrt{z} = frac{36}{36} = 1]So, ( sqrt{z} = 1 ). To find ( z ), I need to square both sides of the equation:[z = (1)^2 = 1]Wait a second, that seems straightforward. So, when ( y = 6 ), ( z = 1 ). Let me just double-check my steps to make sure I didn't make a mistake.1. I started with the inverse variation: ( y^2 cdot sqrt{z} = k ).2. Plugged in ( y = 3 ) and ( z = 16 ) to find ( k ): - ( 3^2 = 9 ) - ( sqrt{16} = 4 ) - ( 9 times 4 = 36 ), so ( k = 36 ).3. Then, used ( y = 6 ) and ( k = 36 ) to find ( z ): - ( 6^2 = 36 ) - Plugged into the equation: ( 36 times sqrt{z} = 36 ) - Divided both sides by 36: ( sqrt{z} = 1 ) - Squared both sides: ( z = 1 )Everything seems to check out. I don't see any errors in my calculations or reasoning. So, I think the answer is correct.Just to think about it another way, since ( y^2 ) varies inversely with ( sqrt{z} ), increasing ( y ) should decrease ( z ). In this case, ( y ) doubled from 3 to 6, so ( y^2 ) went from 9 to 36, which is a factor of 4. Since it's an inverse relationship, ( sqrt{z} ) should decrease by a factor of 4. Originally, ( sqrt{z} = 4 ), so dividing by 4 gives ( sqrt{z} = 1 ), which means ( z = 1 ). That makes sense.So, yeah, I'm confident that ( z = 1 ) when ( y = 6 ).**Final Answer**The value of ( z ) when ( y = 6 ) is boxed{1}.

answer：Okay, so I have this problem where I'm given that the tangent of alpha is -3, and I need to find the value of two expressions. Let me take it step by step because I'm still getting comfortable with trigonometric identities and manipulations.Starting with the first expression: (3 sin α - 3 cos α) divided by (6 cos α + sin α). Hmm, I remember that when dealing with expressions involving sine and cosine, especially when given the tangent, it might be helpful to express everything in terms of tangent. Since tan α is given as -3, maybe I can rewrite the numerator and denominator in terms of tan α.I recall that tan α is sin α over cos α, so maybe if I divide both the numerator and the denominator by cos α, I can express everything in terms of tan α. Let me try that.So, dividing the numerator by cos α: 3 sin α / cos α - 3 cos α / cos α. That simplifies to 3 tan α - 3. Similarly, dividing the denominator by cos α: 6 cos α / cos α + sin α / cos α, which is 6 + tan α.So now the expression becomes (3 tan α - 3) divided by (6 + tan α). Plugging in tan α = -3, let's compute the numerator and denominator separately.Numerator: 3*(-3) - 3 = -9 - 3 = -12.Denominator: 6 + (-3) = 3.So the expression simplifies to -12 divided by 3, which is -4. Okay, that seems straightforward. I think I did that correctly.Now moving on to the second expression: 1 divided by (sin α cos α + 1 + cos 2α). Hmm, this looks a bit more complicated. I remember there are several identities for cos 2α, so maybe I can use one of those to simplify the expression.The double-angle identity for cosine is cos 2α = cos² α - sin² α. Let me substitute that into the expression.So, the denominator becomes sin α cos α + 1 + cos² α - sin² α. Let me rearrange that a bit: 1 + sin α cos α + cos² α - sin² α.Wait, I also remember that sin² α + cos² α = 1. Maybe I can use that to simplify further. Let me see.If I add sin² α + cos² α to the denominator, I can rewrite the denominator as:1 + sin α cos α + cos² α - sin² α = (sin² α + cos² α) + sin α cos α + cos² α - sin² α.Simplifying that, the sin² α and -sin² α cancel out, and we're left with cos² α + sin α cos α + cos² α. Wait, that doesn't seem right. Let me check my steps again.Wait, no, I think I made a mistake there. Let me try again.Starting with the denominator: sin α cos α + 1 + cos 2α.Using the identity cos 2α = cos² α - sin² α, so substituting that in:sin α cos α + 1 + cos² α - sin² α.Now, I can rewrite 1 as sin² α + cos² α, since that's a fundamental identity. So replacing 1:sin α cos α + sin² α + cos² α + cos² α - sin² α.Now, let's combine like terms. The sin² α and -sin² α cancel each other out. Then, we have sin α cos α + cos² α + cos² α, which simplifies to sin α cos α + 2 cos² α.So the denominator is sin α cos α + 2 cos² α. Therefore, the entire expression becomes 1 divided by (sin α cos α + 2 cos² α).Hmm, maybe I can factor out a cos α from the denominator to simplify further. Let's try that.Factoring cos α from the denominator: cos α (sin α + 2 cos α). So now the expression is 1 divided by [cos α (sin α + 2 cos α)].Alternatively, I could express everything in terms of tan α, similar to how I did in the first problem. Let me try that approach.Since tan α = sin α / cos α, let me divide both the numerator and the denominator by cos² α. Wait, actually, since the denominator is cos α (sin α + 2 cos α), if I divide numerator and denominator by cos² α, the expression becomes:Numerator: 1 / cos² α.Denominator: [cos α (sin α + 2 cos α)] / cos² α = (sin α + 2 cos α) / cos α = tan α + 2.So, the entire expression becomes (1 / cos² α) divided by (tan α + 2). Which is the same as (1 / cos² α) multiplied by (1 / (tan α + 2)).But I also know that 1 + tan² α = 1 / cos² α. So, 1 / cos² α is equal to 1 + tan² α.Therefore, substituting back in, the expression becomes (1 + tan² α) divided by (tan α + 2).Now, plugging in tan α = -3:Numerator: 1 + (-3)^2 = 1 + 9 = 10.Denominator: (-3) + 2 = -1.So the expression simplifies to 10 divided by (-1), which is -10.Wait, that seems a bit odd because the original expression was 1 over something, and now it's negative. Let me double-check my steps to make sure I didn't make a mistake.Starting again, the denominator after substitution was sin α cos α + 2 cos² α. Then I factored out cos α to get cos α (sin α + 2 cos α). Then I divided numerator and denominator by cos² α, which gave me (1 / cos² α) divided by (tan α + 2). Then, using 1 / cos² α = 1 + tan² α, so it's (1 + tan² α) / (tan α + 2). Plugging in tan α = -3, that's (1 + 9) / (-3 + 2) = 10 / (-1) = -10. Hmm, seems correct.But just to be thorough, maybe I can approach it differently. Let me try expressing everything in terms of sin α and cos α without using the identity for cos 2α.Starting with the denominator: sin α cos α + 1 + cos 2α.Alternatively, I could use another identity for cos 2α, such as cos 2α = 2 cos² α - 1. Let me try that.So, substituting cos 2α = 2 cos² α - 1 into the denominator:sin α cos α + 1 + 2 cos² α - 1.Simplifying, the 1 and -1 cancel out, leaving sin α cos α + 2 cos² α, which is the same as before. So, same result.So, regardless of which identity I use for cos 2α, I end up with the same denominator: sin α cos α + 2 cos² α.Therefore, my earlier steps seem correct, and the final value is indeed -10.Wait a second, but the original expression was 1 divided by that denominator. So, if the denominator simplifies to -10, then the entire expression is 1 / (-10) = -1/10. Wait, no, that contradicts my earlier conclusion.Hold on, I think I made a mistake in the substitution. Let me go back.Wait, no, in the first approach, I had:The expression is 1 / [sin α cos α + 1 + cos 2α].After substitution, it became 1 / [sin α cos α + 2 cos² α].Then, I factored out cos α, getting 1 / [cos α (sin α + 2 cos α)].Then, I divided numerator and denominator by cos² α, which gave me (1 / cos² α) / (tan α + 2).But 1 / cos² α is 1 + tan² α, so it becomes (1 + tan² α) / (tan α + 2).Plugging in tan α = -3, that's (1 + 9) / (-3 + 2) = 10 / (-1) = -10.Wait, but the original expression is 1 divided by the denominator, so if the denominator simplifies to -10, then the expression is 1 / (-10) = -1/10. But that's not what I got earlier.Wait, no, I think I confused myself. Let me clarify.When I divided numerator and denominator by cos² α, the expression became:[1 / cos² α] / [ (sin α + 2 cos α) / cos α ].Which is equal to [1 / cos² α] * [cos α / (sin α + 2 cos α)].Which simplifies to [1 / (cos α (sin α + 2 cos α))].Wait, that's the same as before. So, perhaps I should have approached it differently.Alternatively, maybe I can express sin α and cos α in terms of tan α.Given tan α = -3, which is sin α / cos α = -3, so sin α = -3 cos α.Also, since sin² α + cos² α = 1, substituting sin α = -3 cos α:(-3 cos α)^2 + cos² α = 1 => 9 cos² α + cos² α = 1 => 10 cos² α = 1 => cos² α = 1/10 => cos α = ±1/√10.Since tan α is negative, alpha is in either the second or fourth quadrant. Depending on the quadrant, cos α could be positive or negative.But since the problem doesn't specify, I might have to consider both possibilities, but perhaps the expressions will simplify regardless.Let me proceed with cos α = 1/√10 and sin α = -3/√10.So, plugging into the denominator of the second expression:sin α cos α + 1 + cos 2α.First, compute sin α cos α: (-3/√10)(1/√10) = -3/10.Then, 1 is just 1.Now, compute cos 2α. Using the identity cos 2α = 2 cos² α - 1: 2*(1/10) - 1 = 2/10 - 1 = 1/5 - 1 = -4/5.So, adding them up: (-3/10) + 1 + (-4/5).Convert to tenths: (-3/10) + (10/10) + (-8/10) = (-3 + 10 - 8)/10 = (-1)/10.So, the denominator is -1/10, and the entire expression is 1 divided by (-1/10), which is -10.Wait, that's consistent with my earlier result. So, the value is indeed -10.But just to be thorough, let me check with cos α = -1/√10 and sin α = 3/√10.Compute sin α cos α: (3/√10)(-1/√10) = -3/10.1 is still 1.cos 2α: 2 cos² α - 1 = 2*(1/10) - 1 = 2/10 - 1 = -4/5.Adding them up: (-3/10) + 1 + (-4/5) = same as before, (-3 + 10 - 8)/10 = (-1)/10.So, denominator is -1/10, and 1 divided by that is -10. So regardless of the quadrant, the value is -10.Therefore, both expressions evaluate to -4 and -10 respectively.I think I've covered all the steps and checked my work to ensure there are no mistakes. It's a bit involved, but breaking it down step by step helps prevent errors.

answer：Okay, so I have this function f(x) defined piecewise. For x less than or equal to 7, it's 2x minus 10, and for x greater than 7, it's 1 over f(x minus 2). I need to find the sum of the first 50 terms of the sequence a_n, where each a_n is f(n) and n is a positive integer. Hmm, okay, let's break this down step by step.First, let me understand the function f(x). For x ≤ 7, it's straightforward: f(x) = 2x - 10. That part is easy. So, for n = 1, 2, 3, up to 7, a_n will just be 2n - 10. Let me compute those values to get a sense of the sequence.Calculating the first few terms:- a_1 = f(1) = 2*1 - 10 = 2 - 10 = -8- a_2 = f(2) = 2*2 - 10 = 4 - 10 = -6- a_3 = f(3) = 2*3 - 10 = 6 - 10 = -4- a_4 = f(4) = 2*4 - 10 = 8 - 10 = -2- a_5 = f(5) = 2*5 - 10 = 10 - 10 = 0- a_6 = f(6) = 2*6 - 10 = 12 - 10 = 2- a_7 = f(7) = 2*7 - 10 = 14 - 10 = 4Okay, so up to n=7, the terms are: -8, -6, -4, -2, 0, 2, 4. That's an arithmetic sequence with a common difference of 2, starting from -8.Now, for n > 7, the function is defined as 1 over f(n-2). So, let's compute the next few terms to see if there's a pattern.- a_8 = f(8) = 1 / f(6) = 1 / 2 = 0.5- a_9 = f(9) = 1 / f(7) = 1 / 4 = 0.25- a_10 = f(10) = 1 / f(8) = 1 / (1/2) = 2- a_11 = f(11) = 1 / f(9) = 1 / (1/4) = 4- a_12 = f(12) = 1 / f(10) = 1 / 2 = 0.5- a_13 = f(13) = 1 / f(11) = 1 / 4 = 0.25- a_14 = f(14) = 1 / f(12) = 1 / (1/2) = 2- a_15 = f(15) = 1 / f(13) = 1 / (1/4) = 4Wait a second, this seems to be repeating a pattern. Let me list out the terms from n=8 onwards:- a_8 = 0.5- a_9 = 0.25- a_10 = 2- a_11 = 4- a_12 = 0.5- a_13 = 0.25- a_14 = 2- a_15 = 4So, starting from n=8, the sequence goes 0.5, 0.25, 2, 4, and then repeats. That's a cycle of 4 terms: 0.5, 0.25, 2, 4. So, every 4 terms after n=7, it repeats.Let me confirm this pattern. For n=16, which is 12 + 4, a_16 should be f(16) = 1 / f(14) = 1 / 2 = 0.5, which matches the pattern. Similarly, a_17 = 1 / f(15) = 1 / 4 = 0.25, and so on. Yep, the cycle is consistent.So, to recap:- The first 7 terms (n=1 to n=7) form an arithmetic sequence: -8, -6, -4, -2, 0, 2, 4.- Starting from n=8, the sequence cycles every 4 terms: 0.5, 0.25, 2, 4.Now, I need to find the sum of the first 50 terms. Let's break this down:1. Sum of the first 7 terms.2. Sum of the remaining 43 terms (from n=8 to n=50). Since the sequence cycles every 4 terms, I can find how many complete cycles there are in 43 terms and the sum of the remaining terms.First, let's compute the sum of the first 7 terms.The first 7 terms are: -8, -6, -4, -2, 0, 2, 4.This is an arithmetic sequence with first term a = -8, common difference d = 2, and number of terms n = 7.The formula for the sum of an arithmetic sequence is S = n/2 * [2a + (n - 1)d].Plugging in the values:S = 7/2 * [2*(-8) + (7 - 1)*2] = 7/2 * [-16 + 12] = 7/2 * (-4) = 7 * (-2) = -14.So, the sum of the first 7 terms is -14.Now, moving on to the next part: the sum from n=8 to n=50, which is 43 terms.Since the sequence from n=8 onwards cycles every 4 terms, let's find how many complete cycles there are in 43 terms and how many extra terms are left.Dividing 43 by 4: 43 ÷ 4 = 10 with a remainder of 3. So, there are 10 complete cycles and 3 extra terms.First, let's compute the sum of one complete cycle (4 terms): 0.5, 0.25, 2, 4.Sum = 0.5 + 0.25 + 2 + 4 = 0.75 + 6 = 6.75.Alternatively, in fractions, 0.5 is 1/2, 0.25 is 1/4, so:Sum = 1/2 + 1/4 + 2 + 4 = (2/4 + 1/4) + 6 = 3/4 + 6 = 6.75 or 27/4.So, each cycle sums to 27/4.Now, 10 complete cycles would sum to 10 * 27/4 = 270/4 = 135/2.Next, we have 3 extra terms. The cycle is 0.5, 0.25, 2, 4. So, the first 3 terms of the cycle are 0.5, 0.25, 2.Sum of these 3 terms: 0.5 + 0.25 + 2 = 0.75 + 2 = 2.75 or 11/4.So, the total sum from n=8 to n=50 is 135/2 + 11/4.To add these, convert 135/2 to 270/4, so 270/4 + 11/4 = 281/4.Now, the total sum of the first 50 terms is the sum of the first 7 terms plus the sum from n=8 to n=50.That is, -14 + 281/4.Convert -14 to quarters: -14 = -56/4.So, total sum = (-56/4) + (281/4) = (281 - 56)/4 = 225/4.225 divided by 4 is 56.25, but since the question probably expects a fraction, 225/4 is the answer.Wait, let me double-check my calculations to make sure I didn't make any mistakes.First, the sum of the first 7 terms: arithmetic sequence with a = -8, d = 2, n =7.Sum = 7/2 * [2*(-8) + 6*2] = 7/2 * (-16 + 12) = 7/2 * (-4) = -14. That seems correct.Sum from n=8 to n=50: 43 terms. 43 divided by 4 is 10 cycles with 3 remaining terms. Each cycle sum is 6.75 or 27/4. 10 cycles: 270/4. 3 extra terms: 0.5 + 0.25 + 2 = 2.75 or 11/4. Total sum: 270/4 + 11/4 = 281/4.Total sum: -14 + 281/4. Convert -14 to -56/4, so total is (-56 + 281)/4 = 225/4. Yep, that's correct.So, the sum of the first 50 terms is 225/4.**Final Answer**The sum of the first 50 terms of the sequence is boxed{dfrac{225}{4}}.

answer：First, I need to determine the smallest prime number that divides the sum (2^{14} + 3^{15}). 1. **Check if the sum is even or odd**: - (2^{14}) is even because any power of 2 is even. - (3^{15}) is odd because any power of 3 is odd. - The sum of an even number and an odd number is odd. Therefore, (2^{14} + 3^{15}) is odd.2. **Check divisibility by 2**: - Since the sum is odd, it is not divisible by 2.3. **Check divisibility by 3**: - Calculate (2^{14} mod 3): - (2 mod 3 = 2) - (2^2 mod 3 = 4 mod 3 = 1) - (2^{14} = (2^2)^7 mod 3 = 1^7 mod 3 = 1) - Therefore, (2^{14} mod 3 = 1). - (3^{15} mod 3 = 0) because any power of 3 is divisible by 3. - Sum modulo 3: (1 + 0 = 1) - Since the sum modulo 3 is 1, it is not divisible by 3.4. **Check divisibility by 5**: - Calculate (2^{14} mod 5): - (2^4 mod 5 = 16 mod 5 = 1) - (2^{14} = 2^{4 times 3 + 2} = (2^4)^3 times 2^2 mod 5 = 1^3 times 4 = 4) - Calculate (3^{15} mod 5): - (3^4 mod 5 = 81 mod 5 = 1) - (3^{15} = 3^{4 times 3 + 3} = (3^4)^3 times 3^3 mod 5 = 1^3 times 27 mod 5 = 2) - Sum modulo 5: (4 + 2 = 6 mod 5 = 1) - Since the sum modulo 5 is 1, it is not divisible by 5.5. **Conclusion**: - The sum (2^{14} + 3^{15}) is not divisible by 2, 3, or 5. Therefore, the smallest prime number dividing the sum is not among the options provided.

answer：Okay, so I have two problems to solve here. Let me take them one at a time. Starting with the first problem: It's about finding the standard equation of an ellipse. The foci are on the x-axis, the focal distance is 4, and it passes through the point M(3, -2√6). Hmm, okay.First, I remember that the standard form of an ellipse with foci on the x-axis is (x²/a²) + (y²/b²) = 1, where a > b. The distance between the foci is 2c, so if the focal distance is 4, that means 2c = 4, so c = 2. So, the foci are at (-2, 0) and (2, 0). Now, I need to find a and b. I know that for an ellipse, the relationship between a, b, and c is a² = b² + c². So, once I find a, I can find b.But wait, how do I find a? The ellipse passes through the point M(3, -2√6). So, I can plug this point into the ellipse equation to get an equation involving a and b.Let me do that. Plugging in x = 3 and y = -2√6 into the equation:(3²)/a² + [(-2√6)²]/b² = 1Calculating that:9/a² + (4*6)/b² = 1Which simplifies to:9/a² + 24/b² = 1But I also know that a² = b² + c², and c = 2, so c² = 4. Therefore, a² = b² + 4.So, I can substitute a² in the equation above with b² + 4:9/(b² + 4) + 24/b² = 1Hmm, that looks a bit complicated, but maybe I can solve for b². Let me set b² = k for simplicity. Then, the equation becomes:9/(k + 4) + 24/k = 1Let me find a common denominator for the left side. The common denominator would be k(k + 4). So:[9k + 24(k + 4)] / [k(k + 4)] = 1Expanding the numerator:9k + 24k + 96 = 33k + 96So, the equation is:(33k + 96) / [k(k + 4)] = 1Multiplying both sides by k(k + 4):33k + 96 = k² + 4kBringing all terms to one side:k² + 4k - 33k - 96 = 0Simplify:k² - 29k - 96 = 0Now, I need to solve this quadratic equation for k. Let me use the quadratic formula:k = [29 ± √(29² + 4*96)] / 2Calculating discriminant:29² = 8414*96 = 384So, discriminant = 841 + 384 = 1225√1225 = 35So, k = [29 ± 35]/2So, two solutions:k = (29 + 35)/2 = 64/2 = 32k = (29 - 35)/2 = (-6)/2 = -3But k = b² can't be negative, so k = 32. Therefore, b² = 32.Then, a² = b² + 4 = 32 + 4 = 36So, a² = 36 and b² = 32. Therefore, the standard equation of the ellipse is:x²/36 + y²/32 = 1Okay, that seems reasonable. Let me double-check by plugging the point M(3, -2√6) into this equation:(3²)/36 + [(-2√6)²]/32 = 9/36 + 24/32 = 1/4 + 3/4 = 1Yes, that works. So, part (1) is done.Moving on to problem (2): Find the standard equation of the parabola with the left directrix of the hyperbola x² - y²/3 = 1 as its directrix and the origin as its vertex.Alright, first, I need to find the left directrix of the hyperbola x² - y²/3 = 1.I recall that for a hyperbola of the form x²/a² - y²/b² = 1, the directrices are at x = ±a²/c, where c is the distance from the center to the foci, and c² = a² + b².So, for this hyperbola, a² = 1, so a = 1. b² = 3, so b = √3.Calculating c² = a² + b² = 1 + 3 = 4, so c = 2.Therefore, the directrices are at x = ±a²/c = ±1/2.So, the left directrix is x = -1/2.Now, the problem says that this left directrix is the directrix of the parabola, and the vertex of the parabola is at the origin (0,0).I need to find the standard equation of this parabola.I remember that for a parabola, the standard form when it opens to the right or left is y² = 4px, where p is the distance from the vertex to the focus (and also from the vertex to the directrix, but on the opposite side).Since the directrix is x = -1/2, and the vertex is at (0,0), the parabola opens to the right because the directrix is to the left of the vertex.So, the standard form should be y² = 4px.But wait, in this case, the directrix is x = -p, right? Because for the standard parabola y² = 4px, the directrix is x = -p.Given that the directrix is x = -1/2, so -p = -1/2, which means p = 1/2.Therefore, the equation of the parabola is y² = 4*(1/2)*x = 2x.So, the standard equation is y² = 2x.Let me just verify this. The vertex is at (0,0), the focus should be at (p,0) = (1/2, 0), and the directrix is x = -1/2. That matches the given condition. So, yes, that seems correct.**Final Answer**(1) The standard equation of the ellipse is boxed{dfrac{x^2}{36} + dfrac{y^2}{32} = 1}.(2) The standard equation of the parabola is boxed{y^2 = 2x}.