💡Alright, so I've got this problem here: Let (a, b, c,) and (d) be real numbers such that (a^{2} + b^{2} + c^{2} + d^{2} = 4). I need to prove that the inequality ((a+2)(b+2) geq c d) holds, and also provide four numbers (a, b, c,) and (d) for which equality holds. Hmm, okay, let's try to break this down step by step.First, I should probably expand the left-hand side of the inequality to see what I'm dealing with. So, ((a+2)(b+2)) expands to (ab + 2a + 2b + 4). So, the inequality becomes:[ab + 2a + 2b + 4 geq c d]Now, I know that (a^2 + b^2 + c^2 + d^2 = 4). Maybe I can use this condition somehow. Let me think about how to relate these two expressions.Perhaps I can rearrange the inequality to incorporate the given condition. If I move all terms to one side, I get:[ab + 2a + 2b + 4 - c d geq 0]But I'm not sure if that helps directly. Maybe I can use some inequality techniques here, like Cauchy-Schwarz or AM-GM. Let me consider Cauchy-Schwarz because it often helps with expressions involving sums and products.Wait, but before that, maybe completing the square could be useful. Let me see. If I look at the terms involving (a) and (b), I have (ab + 2a + 2b). Maybe I can complete the square for (a) and (b).Let me try that. Let's consider (ab + 2a + 2b). Hmm, this looks a bit like part of a square. If I think about ((a + b)^2), that's (a^2 + 2ab + b^2). But I have (ab + 2a + 2b). Maybe I can manipulate it to look like a square.Alternatively, maybe I can use the condition (a^2 + b^2 + c^2 + d^2 = 4) to bound some of these terms. For example, since all squares are non-negative, each of (a^2, b^2, c^2, d^2) is at most 4.But I'm not sure if that's directly helpful. Let me think differently. Maybe I can use the AM-GM inequality on (c) and (d). The AM-GM inequality says that for non-negative real numbers, the arithmetic mean is at least the geometric mean. But (c) and (d) can be negative, so that might complicate things.Wait, but if I consider (c d), the product, it can be positive or negative. However, the left-hand side of the inequality is (ab + 2a + 2b + 4), which might be positive or negative as well. Hmm, this is getting a bit tricky.Let me try another approach. Maybe I can use the Cauchy-Schwarz inequality on the terms involving (c) and (d). The Cauchy-Schwarz inequality states that for any real numbers, ((c^2 + d^2)(1 + 1) geq (c + d)^2). So, (2(c^2 + d^2) geq (c + d)^2). But I'm not sure if that helps with (c d).Alternatively, maybe I can express (c d) in terms of something else. Let me recall that (c d leq frac{c^2 + d^2}{2}) by AM-GM, but since (c) and (d) can be negative, this might not hold. Wait, actually, (c d leq frac{c^2 + d^2}{2}) is always true because (c d leq |c d| leq frac{c^2 + d^2}{2}). So, maybe I can use that.Given that (c d leq frac{c^2 + d^2}{2}), I can substitute this into the inequality:[ab + 2a + 2b + 4 geq frac{c^2 + d^2}{2}]But since (a^2 + b^2 + c^2 + d^2 = 4), we can write (c^2 + d^2 = 4 - a^2 - b^2). Substituting that in, we get:[ab + 2a + 2b + 4 geq frac{4 - a^2 - b^2}{2}]Simplifying the right-hand side:[frac{4 - a^2 - b^2}{2} = 2 - frac{a^2}{2} - frac{b^2}{2}]So, the inequality becomes:[ab + 2a + 2b + 4 geq 2 - frac{a^2}{2} - frac{b^2}{2}]Let's move all terms to the left-hand side:[ab + 2a + 2b + 4 - 2 + frac{a^2}{2} + frac{b^2}{2} geq 0]Simplifying:[ab + 2a + 2b + 2 + frac{a^2}{2} + frac{b^2}{2} geq 0]Hmm, this looks a bit messy. Maybe I can factor this expression or complete the square. Let's see. Let me group the terms involving (a) and (b):[frac{a^2}{2} + ab + 2a + frac{b^2}{2} + 2b + 2]This looks like a quadratic in terms of (a) and (b). Maybe I can write it as:[frac{1}{2}(a^2 + 2ab + b^2) + 2a + 2b + 2]Which simplifies to:[frac{1}{2}(a + b)^2 + 2(a + b) + 2]Let me set (x = a + b). Then the expression becomes:[frac{1}{2}x^2 + 2x + 2]Now, this is a quadratic in (x). Let's see if it's always non-negative. The quadratic is (frac{1}{2}x^2 + 2x + 2). The discriminant is (D = (2)^2 - 4 cdot frac{1}{2} cdot 2 = 4 - 4 = 0). So, the quadratic has a double root at (x = frac{-2}{2 cdot frac{1}{2}} = -2). Therefore, the quadratic can be written as:[frac{1}{2}(x + 2)^2]Which is always non-negative, and equals zero only when (x = -2). So, going back to (x = a + b), we have:[frac{1}{2}(a + b + 2)^2 geq 0]Which is always true, and equals zero when (a + b = -2).So, putting it all together, we've shown that:[(a+2)(b+2) geq c d]And equality holds when (a + b = -2) and (c = d). Now, we need to find specific values of (a, b, c,) and (d) that satisfy both (a + b = -2) and (a^2 + b^2 + c^2 + d^2 = 4), as well as (c = d).Let me choose (a = b = -1). Then (a + b = -2), which satisfies the first condition. Now, since (c = d), let me set (c = d = k). Then, the condition (a^2 + b^2 + c^2 + d^2 = 4) becomes:[(-1)^2 + (-1)^2 + k^2 + k^2 = 1 + 1 + 2k^2 = 2 + 2k^2 = 4]Solving for (k^2):[2k^2 = 2 implies k^2 = 1 implies k = pm 1]So, (c = d = 1) or (c = d = -1). Let's check both cases.First, (c = d = 1):Check the equality condition:[(a+2)(b+2) = (-1 + 2)(-1 + 2) = (1)(1) = 1]And (c d = 1 cdot 1 = 1). So, equality holds.Now, (c = d = -1):[(a+2)(b+2) = (-1 + 2)(-1 + 2) = 1 cdot 1 = 1]And (c d = (-1)(-1) = 1). So, equality also holds here.Therefore, both sets of values satisfy the equality condition. However, let's verify the original condition (a^2 + b^2 + c^2 + d^2 = 4) for both cases.For (c = d = 1):[(-1)^2 + (-1)^2 + 1^2 + 1^2 = 1 + 1 + 1 + 1 = 4]For (c = d = -1):[(-1)^2 + (-1)^2 + (-1)^2 + (-1)^2 = 1 + 1 + 1 + 1 = 4]Both satisfy the condition. So, either set of values works. However, the problem asks to provide four numbers for which equality holds, so I can choose either one. Let's go with (a = -1, b = -1, c = 1, d = 1) as the example.Let me just recap what I did:1. Expanded the left-hand side of the inequality.2. Used the given condition to relate (c d) to (c^2 + d^2).3. Completed the square for the expression involving (a) and (b).4. Showed that the resulting expression is always non-negative.5. Found specific values of (a, b, c,) and (d) that satisfy the equality condition.I think this approach makes sense. I didn't see any mistakes in my reasoning, but let me double-check a few steps.First, when I used the AM-GM inequality on (c d), I considered that (c d leq frac{c^2 + d^2}{2}), which is correct because (c d leq |c d| leq frac{c^2 + d^2}{2}) by the AM-GM inequality.Then, when I substituted (c^2 + d^2 = 4 - a^2 - b^2), that's valid because (a^2 + b^2 + c^2 + d^2 = 4).Completing the square for (a) and (b) was a bit tricky, but I think I did it correctly. I grouped the terms and set (x = a + b), which allowed me to write the expression as a perfect square.Finally, choosing (a = b = -1) and (c = d = 1) satisfies both the equality condition and the given constraint (a^2 + b^2 + c^2 + d^2 = 4). So, that seems correct.I think I've covered all the necessary steps and verified the key points. So, I'm confident that the inequality holds and that the provided values satisfy equality.

💡First, determine the number of cookies Alice can make per cup of flour. She makes 24 cookies with 4 cups, so divide 24 by 4 to get 6 cookies per cup.Next, calculate how many cookies she can make with 5 cups of flour by multiplying the number of cookies per cup (6) by the number of cups (5). This gives 30 cookies.

💡Okay, so I have this problem about an ellipse and a circle, and I need to figure out which statements are true. Let's break it down step by step.First, the ellipse is given by the equation ( C: frac{x^2}{4} + y^2 = 1 ). I remember that the standard form of an ellipse is ( frac{x^2}{a^2} + frac{y^2}{b^2} = 1 ), where ( a ) is the semi-major axis and ( b ) is the semi-minor axis. In this case, ( a^2 = 4 ) so ( a = 2 ), and ( b^2 = 1 ) so ( b = 1 ).Now, the first statement, A, says the focal length of the ellipse ( C ) is ( sqrt{3} ). I recall that the distance from the center to each focus (c) is given by ( c = sqrt{a^2 - b^2} ). Plugging in the values, ( c = sqrt{4 - 1} = sqrt{3} ). But wait, the focal length is the distance between the two foci, which would be ( 2c ), so that's ( 2sqrt{3} ). So statement A is incorrect because it says the focal length is ( sqrt{3} ), not ( 2sqrt{3} ).Moving on to statement B: The eccentricity of the ellipse ( C ) is ( frac{sqrt{3}}{2} ). Eccentricity ( e ) is calculated as ( e = frac{c}{a} ). We already found ( c = sqrt{3} ) and ( a = 2 ), so ( e = frac{sqrt{3}}{2} ). That matches statement B, so B is correct.Next, statement C: The circle ( D ) lies inside the ellipse ( C ). The circle is given by ( D: (x + 1)^2 + y^2 = frac{1}{4} ). This is a circle with center at ( (-1, 0) ) and radius ( frac{1}{2} ). To check if it lies inside the ellipse, I can see if the entire circle is within the ellipse. The ellipse extends from ( x = -2 ) to ( x = 2 ) and ( y = -1 ) to ( y = 1 ). The circle is centered at ( (-1, 0) ) with a small radius, so it should be entirely within the ellipse. To be thorough, I can plug the center of the circle into the ellipse equation: ( frac{(-1)^2}{4} + 0^2 = frac{1}{4} ), which is less than 1, so the center is inside the ellipse. Since the radius is small, the entire circle should be inside. Therefore, statement C is correct.Finally, statement D: The minimum value of ( |PQ| ) is ( frac{sqrt{6}}{3} ). Here, ( P ) is on the ellipse and ( Q ) is on the circle. To find the minimum distance between any point on the ellipse and any point on the circle, I need to consider the distance between the two curves. The distance between two points ( P(x_1, y_1) ) and ( Q(x_2, y_2) ) is ( sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} ). To minimize this, I might consider the distance from the ellipse to the center of the circle and then subtract the radius. The center of the circle is ( (-1, 0) ), so the distance from a point ( (x, y) ) on the ellipse to ( (-1, 0) ) is ( sqrt{(x + 1)^2 + y^2} ). To minimize this, I can set up the expression ( (x + 1)^2 + y^2 ) and use the ellipse equation to substitute ( y^2 = 1 - frac{x^2}{4} ). So the expression becomes ( (x + 1)^2 + 1 - frac{x^2}{4} ). Expanding and simplifying: ( x^2 + 2x + 1 + 1 - frac{x^2}{4} = frac{3x^2}{4} + 2x + 2 ). To find the minimum, take the derivative with respect to ( x ): ( frac{d}{dx} (frac{3x^2}{4} + 2x + 2) = frac{3x}{2} + 2 ). Setting this equal to zero: ( frac{3x}{2} + 2 = 0 ) leads to ( x = -frac{4}{3} ). Plugging this back into the expression: ( frac{3(-frac{4}{3})^2}{4} + 2(-frac{4}{3}) + 2 = frac{3 cdot frac{16}{9}}{4} - frac{8}{3} + 2 = frac{4}{3} - frac{8}{3} + 2 = -frac{4}{3} + 2 = frac{2}{3} ). So the minimum distance squared is ( frac{2}{3} ), so the distance is ( sqrt{frac{2}{3}} = frac{sqrt{6}}{3} ). However, this is the distance from the ellipse to the center of the circle. Since the radius of the circle is ( frac{1}{2} ), the minimum distance between any point on the ellipse and any point on the circle would be ( frac{sqrt{6}}{3} - frac{1}{2} ). Calculating this: ( frac{sqrt{6}}{3} approx 0.816 ) and ( frac{1}{2} = 0.5 ), so ( 0.816 - 0.5 = 0.316 ). But ( frac{sqrt{6}}{3} approx 0.816 ), which is larger than 0.316, so I think I made a mistake here. Maybe I shouldn't subtract the radius because the minimum distance could be achieved at a point where the line connecting the ellipse and the circle's center intersects the circle. Alternatively, perhaps the minimum distance is indeed ( frac{sqrt{6}}{3} ) because the closest point on the circle to the ellipse is in the direction towards the center. I'm a bit confused here, but I'll go with the calculation that the minimum distance is ( frac{sqrt{6}}{3} ), so statement D is correct.Wait, I think I need to reconsider. The minimum distance between the ellipse and the circle would be the distance from the ellipse to the center of the circle minus the radius of the circle. So if the minimum distance from the ellipse to the center is ( frac{sqrt{6}}{3} ), then the minimum distance to the circle would be ( frac{sqrt{6}}{3} - frac{1}{2} ). Let's compute ( frac{sqrt{6}}{3} approx 0.816 ) and ( frac{1}{2} = 0.5 ), so ( 0.816 - 0.5 = 0.316 ). But ( 0.316 ) is approximately ( frac{sqrt{6}}{3} - frac{1}{2} ), which is about ( 0.316 ). However, ( frac{sqrt{6}}{3} ) is about ( 0.816 ), which is larger than ( 0.316 ), so I think I might have made a mistake in interpreting the minimum distance. Maybe the minimum distance between the two curves is indeed ( frac{sqrt{6}}{3} ), considering the closest points. Alternatively, perhaps the minimum distance is achieved when the line connecting the points ( P ) and ( Q ) passes through the center of the circle. I'm not entirely sure, but given the calculations, I'll tentatively say that statement D is correct.Wait, another approach: parametrize the ellipse and the circle and find the minimum distance. The ellipse can be parametrized as ( x = 2costheta ), ( y = sintheta ). The circle can be parametrized as ( x = -1 + frac{1}{2}cosphi ), ( y = frac{1}{2}sinphi ). The distance squared between ( P ) and ( Q ) is ( (2costheta + 1 - frac{1}{2}cosphi)^2 + (sintheta - frac{1}{2}sinphi)^2 ). To find the minimum, we'd need to minimize this expression over ( theta ) and ( phi ), which seems complicated. Alternatively, consider the distance from the ellipse to the circle's center and subtract the radius. If the minimum distance from the ellipse to the center is ( frac{sqrt{6}}{3} ), then the minimum distance to the circle is ( frac{sqrt{6}}{3} - frac{1}{2} ). But I'm not sure if this is the correct approach because the closest point on the circle might not lie along the line connecting the ellipse point to the center. Maybe I should use calculus to find the minimum distance. Alternatively, perhaps the minimum distance is indeed ( frac{sqrt{6}}{3} ) as calculated earlier, considering the closest approach. I'm a bit uncertain, but I'll go with statement D being correct.So, summarizing:- A: Incorrect- B: Correct- C: Correct- D: CorrectBut wait, earlier I thought about subtracting the radius, which would make the minimum distance less than ( frac{sqrt{6}}{3} ). Maybe I was wrong about D. Let me double-check. If the minimum distance from the ellipse to the center is ( frac{sqrt{6}}{3} ), and the radius is ( frac{1}{2} ), then the minimum distance between the ellipse and the circle would be ( frac{sqrt{6}}{3} - frac{1}{2} ). Calculating this: ( frac{sqrt{6}}{3} approx 0.816 ), ( frac{1}{2} = 0.5 ), so ( 0.816 - 0.5 = 0.316 ). But ( 0.316 ) is approximately ( frac{sqrt{6}}{3} - frac{1}{2} ), which is about ( 0.316 ). However, ( frac{sqrt{6}}{3} ) is about ( 0.816 ), which is larger than ( 0.316 ), so I think I made a mistake in interpreting the minimum distance. Maybe the minimum distance between the two curves is indeed ( frac{sqrt{6}}{3} ), considering the closest points. Alternatively, perhaps the minimum distance is achieved when the line connecting the points ( P ) and ( Q ) passes through the center of the circle. I'm not entirely sure, but given the calculations, I'll tentatively say that statement D is correct.Wait, another thought: the minimum distance between two curves is the smallest distance between any two points on the curves. So, if the closest point on the ellipse to the center of the circle is at a distance ( frac{sqrt{6}}{3} ), and the radius of the circle is ( frac{1}{2} ), then the minimum distance between the ellipse and the circle would be ( frac{sqrt{6}}{3} - frac{1}{2} ). Let's compute this: ( frac{sqrt{6}}{3} approx 0.816 ), ( frac{1}{2} = 0.5 ), so ( 0.816 - 0.5 = 0.316 ). But ( 0.316 ) is approximately ( frac{sqrt{6}}{3} - frac{1}{2} ), which is about ( 0.316 ). However, ( frac{sqrt{6}}{3} ) is about ( 0.816 ), which is larger than ( 0.316 ), so I think I might have made a mistake in interpreting the minimum distance. Maybe the minimum distance between the two curves is indeed ( frac{sqrt{6}}{3} ), considering the closest approach. Alternatively, perhaps the minimum distance is achieved when the line connecting the points ( P ) and ( Q ) passes through the center of the circle. I'm not entirely sure, but I'll go with the calculation that the minimum distance is ( frac{sqrt{6}}{3} ), so statement D is correct.Wait, I think I need to correct myself. The minimum distance from the ellipse to the center of the circle is ( frac{sqrt{6}}{3} ), but the minimum distance from the ellipse to the circle would be this distance minus the radius of the circle, which is ( frac{sqrt{6}}{3} - frac{1}{2} ). Let's compute this:( frac{sqrt{6}}{3} approx 0.816 )( frac{1}{2} = 0.5 )So, ( 0.816 - 0.5 = 0.316 )But ( 0.316 ) is approximately ( frac{sqrt{6}}{3} - frac{1}{2} ), which is about ( 0.316 ). However, ( frac{sqrt{6}}{3} ) is about ( 0.816 ), which is larger than ( 0.316 ), so I think I made a mistake in interpreting the minimum distance. The minimum distance between the two curves should be the distance from the ellipse to the circle's center minus the circle's radius, which is ( frac{sqrt{6}}{3} - frac{1}{2} approx 0.316 ). But statement D says the minimum value is ( frac{sqrt{6}}{3} approx 0.816 ), which is larger than the actual minimum distance. Therefore, statement D is incorrect.Wait, now I'm really confused. Let me try to clarify:1. The distance from a point on the ellipse to the center of the circle is ( frac{sqrt{6}}{3} ).2. The circle has a radius of ( frac{1}{2} ).3. The minimum distance between the ellipse and the circle would be the distance from the ellipse to the center minus the radius, assuming the closest point on the circle lies along the line connecting the ellipse point to the center.So, ( text{Minimum distance} = frac{sqrt{6}}{3} - frac{1}{2} approx 0.816 - 0.5 = 0.316 ).But statement D says the minimum distance is ( frac{sqrt{6}}{3} approx 0.816 ), which is actually the distance from the ellipse to the center, not to the circle itself. Therefore, statement D is incorrect.So, correcting myself, statement D is incorrect because the minimum distance is less than ( frac{sqrt{6}}{3} ).Therefore, the correct statements are B and C.

💡Okay, so I have this problem here about triangle ABC. It says that AB is 30 cm, AC is 15 cm, and angle A is 90 degrees. I need to find the area of triangle ABC. Hmm, let me think about how to approach this.First, I remember that the area of a triangle is generally calculated using the formula: (base * height) / 2. But wait, in this case, since it's a right-angled triangle at A, that should make things easier, right? Because in a right-angled triangle, the two sides forming the right angle can be considered the base and the height.So, looking at the triangle, point A is the right angle, and the sides AB and AC are connected to it. That means AB and AC are the two legs of the triangle, and BC would be the hypotenuse. So, AB is 30 cm, and AC is 15 cm. Therefore, AB can be the base, and AC can be the height.Let me write down the formula again to make sure I get it right. The area of a triangle is (base * height) / 2. So, plugging in the values I have: base is 30 cm, height is 15 cm. So, the area should be (30 * 15) / 2.Wait, let me compute that. 30 multiplied by 15 is 450. Then, dividing that by 2 gives me 225. So, the area should be 225 cm².But just to double-check, maybe I should visualize the triangle. If AB is 30 cm and AC is 15 cm, and they meet at a right angle at A, then the triangle is a right-angled triangle with legs 30 and 15. So, yes, the area is indeed (30 * 15) / 2, which is 225 cm².Alternatively, I could think about the Pythagorean theorem to find the hypotenuse, but since the problem only asks for the area, I don't think that's necessary here. The area is straightforward with the given sides.Wait, another thought: sometimes, people confuse which sides are the base and height, especially if the triangle isn't drawn to scale. But in this case, since it's a right-angled triangle, the sides forming the right angle are definitely the base and height. So, I don't have to worry about that.Just to recap: right-angled triangle, legs are 30 cm and 15 cm, area is (30 * 15) / 2 = 225 cm². Yep, that seems right.I don't think I made any mistakes here. The calculation seems straightforward, and I double-checked the reasoning. So, I'm confident that the area is 225 cm².

💡First, I need to determine Sam's weight based on the information that Jack weighs 52 pounds, which is 20% less than Sam's weight. This means Jack's weight is 80% of Sam's weight.Next, I'll calculate Lisa's weight, knowing that she weighs 40% more than Jack. This involves adding 40% of Jack's weight to his current weight.Then, I'll find Daisy's weight, which is one-third of the combined weight of Jack and Lisa. This requires summing Jack's and Lisa's weights and then dividing by three.After obtaining all individual weights, I'll sum them up to ensure the total weight matches the given 240 pounds. If there's a discrepancy, I'll check for any calculation errors.Finally, I'll find the difference in weight between Sam and Daisy by subtracting Daisy's weight from Sam's weight.

💡First, I'll define the unknown number as ( x ).According to the problem, I need to add 5 to this number:[ x + 5 ]Next, I'll multiply the sum by 2:[ 2(x + 5) ]Then, I'll divide the product by 5:[ frac{2(x + 5)}{5} ]After that, I'll subtract 5 from the quotient:[ frac{2(x + 5)}{5} - 5 ]The problem states that the remainder left is half of 44, which is:[ frac{44}{2} = 22 ]So, I can set up the equation:[ frac{2(x + 5)}{5} - 5 = 22 ]To solve for ( x ), I'll first isolate the fraction:[ frac{2(x + 5)}{5} = 22 + 5 ][ frac{2(x + 5)}{5} = 27 ]Next, I'll eliminate the denominator by multiplying both sides by 5:[ 2(x + 5) = 135 ]Then, I'll divide both sides by 2 to simplify:[ x + 5 = 67.5 ]Finally, I'll subtract 5 from both sides to find the value of ( x ):[ x = 67.5 - 5 ][ x = 62.5 ]Therefore, the unknown number is 62.5.