💡First, I need to determine how many slices James ate from the first pizza. Since the pizza has 6 slices and James ate 5/6 of it, I can calculate the number of slices he consumed by multiplying 5/6 by 6.Next, I'll subtract the number of slices James ate from the total number of slices to find out how many slices are left on the first pizza.Then, I'll calculate how many slices Lisa ate from the second pizza. Lisa ate 1/3 of the second pizza, so I'll multiply 1/3 by 6 to find the number of slices she consumed.Finally, I'll subtract the number of slices Lisa ate from the total number of slices to determine how many slices remain on the second pizza.

💡Okay, so I have this problem about a store selling clothing, and I need to figure out how much they should reduce the price per piece to make a certain profit. Let me try to understand the problem step by step.First, the store buys each piece of clothing for 45 yuan and sells it for 65 yuan. So, the original profit per piece is 65 minus 45, which is 20 yuan. That makes sense. They sell an average of 30 pieces per day, so their daily profit right now is 20 yuan multiplied by 30, which is 600 yuan. But they want to make more profit—specifically, 800 yuan per day. To achieve this, they're considering reducing the price. The problem says that for every 1 yuan reduction in price, they can sell an additional 5 pieces per day. So, if they reduce the price by 1 yuan, they'll sell 30 plus 5, which is 35 pieces. If they reduce it by 2 yuan, they'll sell 30 plus 10, which is 40 pieces, and so on. Okay, so there's a trade-off here: reducing the price means each piece makes less profit, but they sell more pieces. The goal is to find the right balance where the total profit becomes 800 yuan per day.Let me define some variables to make this clearer. Let's say x is the number of yuan by which the price is reduced. So, the new selling price per piece is 65 minus x yuan. The cost price remains the same at 45 yuan, so the new profit per piece is (65 - x) minus 45, which simplifies to 20 - x yuan.Now, the number of pieces sold per day increases by 5 for each yuan reduced. So, if they reduce the price by x yuan, they'll sell 30 plus 5x pieces per day.The total profit is then the profit per piece multiplied by the number of pieces sold. So, the total profit equation would be:(20 - x) * (30 + 5x) = 800Let me write that out:(20 - x)(30 + 5x) = 800Now, I need to solve this equation for x. First, I'll expand the left side:20 * 30 + 20 * 5x - x * 30 - x * 5x = 800Calculating each term:20 * 30 = 60020 * 5x = 100x-x * 30 = -30x-x * 5x = -5x²So, putting it all together:600 + 100x - 30x - 5x² = 800Combine like terms:600 + 70x - 5x² = 800Now, let's move all terms to one side to set the equation to zero:-5x² + 70x + 600 - 800 = 0Simplify:-5x² + 70x - 200 = 0It's a quadratic equation, and I can simplify it further by dividing all terms by -5 to make the coefficients smaller:x² - 14x + 40 = 0Now, I need to solve this quadratic equation. I can try factoring it. Let's see if there are two numbers that multiply to 40 and add up to -14.Hmm, 40 can be factored into 1 and 40, 2 and 20, 4 and 10, 5 and 8. Looking for a pair that adds up to 14. 10 and 4 add up to 14. But since the middle term is -14x, both numbers should be negative to add up to -14 and multiply to 40.So, the factors would be (x - 10)(x - 4) = 0Setting each factor equal to zero:x - 10 = 0 => x = 10x - 4 = 0 => x = 4So, the solutions are x = 10 and x = 4.Now, I need to interpret these solutions in the context of the problem. If x = 10, that means reducing the price by 10 yuan. The new selling price would be 65 - 10 = 55 yuan, and the number of pieces sold would be 30 + 5*10 = 80 pieces. The profit per piece would be 55 - 45 = 10 yuan, and total profit would be 10 * 80 = 800 yuan, which matches the goal.If x = 4, the price reduction is 4 yuan, so the selling price is 65 - 4 = 61 yuan. The number of pieces sold would be 30 + 5*4 = 50 pieces. The profit per piece is 61 - 45 = 16 yuan, and total profit is 16 * 50 = 800 yuan, which also matches the goal.So, both solutions are mathematically valid. However, the problem mentions that the store wants to reduce inventory quickly. Reducing the price by more would likely lead to selling more pieces, which helps in reducing inventory faster. Therefore, reducing the price by 10 yuan would result in selling 80 pieces per day, which is more than the 50 pieces sold with a 4 yuan reduction.Therefore, the store should reduce the price by 10 yuan per piece to achieve the desired profit and reduce inventory quickly.

💡Alright, so I have this problem where I need to figure out which angles can be constructed using a compass and a straightedge, given an angle α. Hmm, okay, let's start by recalling what I know about compass and straightedge constructions. I remember that with these tools, you can do things like drawing lines, circles, and bisecting angles. But I'm not entirely sure about the specifics when it comes to constructing angles based on a given angle α.First, I think it's important to understand what it means to "construct" an angle. From what I remember, constructing an angle means creating it using only the compass and straightedge, starting from some given elements. In this case, the given element is the angle α. So, I need to figure out what other angles can be made from α using these tools.I recall that one of the fundamental operations with a compass and straightedge is angle bisecting. That means if I have an angle, I can split it into two equal smaller angles. So, if I have angle α, I can create an angle of α/2. Similarly, if I bisect that angle again, I can get α/4, and so on. This suggests that I can create angles that are fractions of α, specifically of the form α/(2^n), where n is a positive integer. That seems straightforward enough.But the problem also mentions constructing angles using a compass and straightedge, given α. So, does that mean I can combine these bisected angles in some way? Maybe I can add or subtract these fractions of α to create other angles. For example, if I have α/2 and α/4, could I add them together to get 3α/4? That seems plausible. In fact, if I can create angles like α/2, α/4, α/8, etc., then by adding or subtracting these, I can create angles that are multiples of these fractions.So, more generally, I think I can create angles that are integer multiples of these fractions. That is, angles of the form k*(α/(2^n)), where k is an integer. This would allow me to create angles like α/2, 3α/4, 5α/8, and so on, depending on the values of k and n.But wait, the problem also mentions angles that can be constructed using just a compass and straightedge without considering α. I think those are the standard constructible angles, like 30 degrees, 45 degrees, 60 degrees, and so on. These angles are special because they can be constructed without any given angle. So, if I can create these standard angles independently, I can also combine them with the angles derived from α.Putting this together, I think the angles that can be constructed using a compass and straightedge, given α, are of the form (k/(2^n))*α plus or minus some standard constructible angle β. This would mean that any angle that can be expressed as a combination of a fraction of α and a standard constructible angle is constructible.To make sure I'm on the right track, let me think about an example. Suppose α is 90 degrees, which is a right angle. Using a compass and straightedge, I can bisect it to get 45 degrees, bisect again to get 22.5 degrees, and so on. Additionally, I can create standard angles like 30 degrees or 60 degrees. So, if I wanted to create, say, 67.5 degrees, I could add 45 degrees and 22.5 degrees. That works out. Similarly, if I wanted to create 15 degrees, I could subtract 30 degrees from 45 degrees. So, this seems to align with the idea of combining fractions of α with standard constructible angles.Another example: if α is 60 degrees, I can bisect it to get 30 degrees, bisect again to get 15 degrees, and so on. I can also create standard angles like 45 degrees. So, combining these, I can create angles like 15 degrees + 45 degrees = 60 degrees, or 30 degrees - 15 degrees = 15 degrees, and so forth. This further supports the idea that the constructible angles are combinations of fractions of α and standard angles.But I should also consider negative angles or angles greater than 360 degrees. Since angles are periodic modulo 360 degrees, any angle can be represented within a 0 to 360-degree range by adding or subtracting multiples of 360 degrees. However, in the context of compass and straightedge constructions, we're typically concerned with angles between 0 and 180 degrees, as angles greater than 180 degrees can be represented as their supplementary angles.So, in summary, given an angle α, the constructible angles using a compass and straightedge would be any angle that can be expressed as a combination of fractions of α (specifically, k/(2^n) times α) and standard constructible angles. This means that the set of constructible angles is quite extensive, as it includes not only the fractions of α but also the ability to combine them with other known constructible angles.I think I've covered the main points, but let me just verify if there are any limitations or exceptions. For instance, not all angles can be constructed with a compass and straightedge, especially those that require trisection of an arbitrary angle, which is famously impossible. However, in this case, since we're starting with a specific angle α and using bisecting operations, we're within the realm of possible constructions.Also, the combination with standard constructible angles β is important because it allows us to adjust the resulting angle to fit within the desired range or to achieve specific angular relationships. This flexibility is key to understanding the full scope of constructible angles given α.Alright, I feel confident that I've thought through this thoroughly. The next step is to present this understanding in a clear and concise manner, ensuring that all the key points are covered and that the reasoning is logical and easy to follow.

💡First, I need to determine the elevation of Boris' mountain. Since Boris' mountain is 2,500 feet shorter than Hugo's mountain, I subtract 2,500 feet from Hugo's mountain elevation:10,000 feet (Hugo's mountain) - 2,500 feet = 7,500 feet (Boris' mountain)Next, I'll let ( x ) represent the number of times Hugo climbed his mountain. Since Boris needs to climb his mountain 4 times to match the elevation Hugo climbed, I can set up the following equation:( 10,000 text{ feet} times x ) (Hugo's total feet climbed) = ( 7,500 text{ feet} times 4 ) (Boris' total feet climbed)Now, I'll solve for ( x ):( 10,000x = 7,500 times 4 )( 10,000x = 30,000 )Dividing both sides by 10,000 to find ( x ):( x = frac{30,000}{10,000} )( x = 3 )Therefore, Hugo climbed his mountain 3 times.

💡Okay, so I have this problem here: "The distance from the focus of the parabola ( y^2 = 20x ) to its directrix is ______." Alright, I need to figure out this distance. Hmm, let me think about what I know regarding parabolas.First, I remember that a parabola is a set of points equidistant from a fixed point called the focus and a fixed line called the directrix. So, for any point on the parabola, its distance to the focus is equal to its distance to the directrix. But in this problem, they're asking for the distance between the focus and the directrix themselves, not the distance from a point on the parabola to these.Alright, so I need to recall the standard form of a parabola and how the focus and directrix are defined in that form. I think the standard form for a parabola that opens to the right or left is ( y^2 = 4px ), where ( p ) is the distance from the vertex to the focus (and also from the vertex to the directrix). So, in this case, our equation is ( y^2 = 20x ). Comparing this to the standard form ( y^2 = 4px ), I can see that ( 4p = 20 ). Let me write that down:( 4p = 20 )So, solving for ( p ):( p = frac{20}{4} = 5 )Okay, so ( p = 5 ). That means the focus is 5 units away from the vertex, and the directrix is also 5 units away from the vertex but on the opposite side. Since the parabola opens to the right (because the equation is ( y^2 = 20x ), which is positive), the focus is at ( (p, 0) ), which would be ( (5, 0) ), and the directrix is the vertical line ( x = -p ), so ( x = -5 ).Now, the question is asking for the distance from the focus to the directrix. Hmm, so if the focus is at ( (5, 0) ) and the directrix is the line ( x = -5 ), how do I find the distance between them?I think the distance from a point to a line is calculated by the perpendicular distance. In this case, since the directrix is a vertical line, the distance from the focus to the directrix would just be the horizontal distance between them. So, the focus is at ( x = 5 ) and the directrix is at ( x = -5 ). The distance between these two vertical lines is the difference in their x-coordinates.So, the distance should be ( 5 - (-5) = 10 ). Wait, is that right? Let me visualize it. If the focus is at 5 on the x-axis and the directrix is at -5, the distance between them is indeed 10 units. That makes sense because the vertex is halfway between the focus and the directrix, so the total distance from focus to directrix is twice the distance from vertex to focus, which is ( 2p ).Let me confirm that. Since ( p = 5 ), then the distance from the vertex to the focus is 5, and the distance from the vertex to the directrix is also 5. So, the total distance between the focus and the directrix is ( 5 + 5 = 10 ). Yep, that seems correct.I think I might have confused myself earlier by thinking it was just ( p ), but no, it's actually ( 2p ). So, in this case, ( 2p = 10 ).Just to make sure I'm not making a mistake, let me recall the formula for the distance from the focus to the directrix in a parabola. I believe it's indeed ( 2p ), where ( p ) is the focal length. So, in our case, since ( p = 5 ), the distance is ( 10 ).Therefore, the distance from the focus of the parabola ( y^2 = 20x ) to its directrix is 10 units.**Final Answer**The distance from the focus of the parabola ( y^2 = 20x ) to its directrix is boxed{10}.

💡First, I need to set up the equation based on the given information. The equation is:[frac{6y}{20} + frac{3y}{d} = 0.60y]Next, I'll simplify (frac{6y}{20}) by reducing the fraction. Dividing both the numerator and the denominator by 2 gives:[frac{3y}{10}]Now, the equation looks like this:[frac{3y}{10} + frac{3y}{d} = 0.60y]To combine the fractions on the left side, I'll find a common denominator, which is (10d). I'll rewrite each fraction with this common denominator:[frac{3y cdot d}{10d} + frac{3y cdot 10}{10d} = 0.60y]Combining the fractions:[frac{3yd + 30y}{10d} = 0.60y]To eliminate the denominator, I'll multiply both sides of the equation by (10d):[3yd + 30y = 0.60y cdot 10d]Simplifying the right side:[3yd + 30y = 6yd]Next, I'll move all terms involving (d) to one side of the equation:[3yd - 6yd = -30y]Combining like terms:[-3yd = -30y]Finally, I'll solve for (d) by dividing both sides by (-3y):[d = frac{-30y}{-3y} = 10]So, the value of (d) is (boxed{10}).