💡Okay, so I have this problem here: "When each edge of a cube is increased by 30%, by what percentage is the surface area of the cube increased?" Hmm, let me try to figure this out step by step.First, I know that a cube has all its edges of equal length. Let's call the original length of each edge 's'. So, the original surface area of the cube would be... well, a cube has 6 faces, right? Each face is a square with area s squared. So, the total surface area is 6 times s squared, which is 6s². Got that down.Now, the problem says each edge is increased by 30%. Okay, so if the original edge length is 's', increasing it by 30% would make the new edge length s plus 30% of s. Let me write that out: new edge length = s + 0.3s. That simplifies to 1.3s. So, the new edge length is 1.3 times the original edge length.Alright, now I need to find the new surface area with this increased edge length. Since the surface area of a cube is 6 times the edge length squared, the new surface area should be 6 times (1.3s) squared. Let me calculate that: (1.3s) squared is 1.69s². So, multiplying that by 6 gives 6 * 1.69s², which is 10.14s². Wait, so the original surface area was 6s², and the new surface area is 10.14s². To find the percentage increase, I think I need to compare the difference between the new and original surface areas to the original surface area. So, the increase in surface area is 10.14s² - 6s², which is 4.14s². To find the percentage increase, I divide this increase by the original surface area and then multiply by 100. That would be (4.14s² / 6s²) * 100. Simplifying that, the s² terms cancel out, so it's (4.14 / 6) * 100. Let me compute 4.14 divided by 6. Hmm, 4 divided by 6 is approximately 0.6667, and 0.14 divided by 6 is about 0.0233. Adding those together gives roughly 0.69. So, 0.69 times 100 is 69%. Therefore, the surface area increases by 69%. Let me just double-check my calculations to make sure I didn't make a mistake. Original edge length 's', surface area 6s². New edge length 1.3s, new surface area 6*(1.3s)² = 6*1.69s² = 10.14s². Difference is 4.14s², which is 4.14/6 = 0.69, so 69%. Yeah, that seems right.I guess another way to think about it is that since surface area depends on the square of the edge length, if the edge length increases by a factor, the surface area increases by the square of that factor. So, increasing by 30% is a factor of 1.3, so the surface area increases by (1.3)² = 1.69, which is a 69% increase. That confirms my earlier result. Okay, I feel confident now that the surface area increases by 69%.**Final Answer**The surface area of the cube is increased by boxed{69%}.

💡First, let ( M ) represent Mickey's age and ( J ) represent Jerry's age.According to the problem, Mickey's age is 6 years less than 200% of Jerry's age. This can be written as:[ M = 2J - 6 ]We are given that Mickey is 18 years old, so:[ 18 = 2J - 6 ]To solve for ( J ), add 6 to both sides of the equation:[ 18 + 6 = 2J ][ 24 = 2J ]Next, divide both sides by 2:[ J = frac{24}{2} ][ J = 12 ]Therefore, Jerry is 12 years old.

💡First, I need to find the zero of the function ( f(x) = x + 1 ). The zero of a function is the value of ( x ) where the function equals zero.So, I set ( f(x) ) equal to zero:[x + 1 = 0]Next, I solve for ( x ) by subtracting 1 from both sides of the equation:[x = -1]Therefore, the zero of the function ( f(x) = x + 1 ) is (-1).

💡Okay, so I have this problem about triangle XYZ formed by the intersections of Pine Road, Oak Road, and a utility line. Pine Road runs north-south, and Oak Road runs east-west. There's also Elm Street, which goes from point X to the midpoint M of YZ. They tell me that YM is 2 miles and MX is 3 miles. The entire distance of YZ along Oak Road is 5 miles. I need to find the area of triangle XMY.Alright, let's break this down. First, I should probably visualize the map. Pine Road is north-south, so it's vertical, and Oak Road is east-west, so it's horizontal. The utility line must be the third side of the triangle, connecting somewhere to form triangle XYZ.Elm Street goes from X to the midpoint M of YZ. So, M is the midpoint, which means YM equals MZ. But wait, they say YM is 2 miles and MX is 3 miles. Hmm, that's interesting because if M is the midpoint, then YM should equal MZ, but MX is given as 3 miles. Maybe MX is the length from X to M, not along YZ.Wait, the entire distance of YZ along Oak Road is 5 miles. So, YZ is 5 miles long. Since M is the midpoint, YM should be half of that, which is 2.5 miles. But the problem says YM is 2 miles. That seems contradictory. Maybe I'm misunderstanding something.Let me read the problem again. It says, "The triangle is further divided by Elm Street running from point X to midpoint M of YZ. YM measures 2 miles and MX measures 3 miles." So, YM is 2 miles, and MX is 3 miles. But if M is the midpoint of YZ, then YM should equal MZ, right? So, if YM is 2 miles, MZ should also be 2 miles, making YZ 4 miles. But the problem says YZ is 5 miles. That doesn't add up.Wait, maybe I'm misinterpreting the distances. YM measures 2 miles, and MX measures 3 miles. So, from Y to M is 2 miles, and from M to X is 3 miles. But M is the midpoint of YZ, so from M to Z should also be 2 miles. That would make YZ 4 miles, but the problem says YZ is 5 miles. There's a discrepancy here.Perhaps the distances YM and MX are not along YZ but in some other direction. Maybe Elm Street is not along YZ but from X to M, which is the midpoint. So, YM is 2 miles along YZ, and MX is 3 miles from M to X. If YZ is 5 miles, then M is the midpoint, so YM should be 2.5 miles, but it's given as 2 miles. That still doesn't make sense.Wait, maybe the problem is that M is not the midpoint in terms of distance but in terms of the line segment YZ. So, if YZ is 5 miles, then M divides YZ into two equal parts, each 2.5 miles. But the problem says YM is 2 miles and MX is 3 miles. So, perhaps MX is not along YZ but from M to X, which is a different direction.So, maybe triangle XYZ has YZ as the base, which is 5 miles, and X is somewhere above YZ. Elm Street goes from X to M, the midpoint of YZ. So, YM is 2.5 miles, but the problem says YM is 2 miles. Hmm, this is confusing.Wait, maybe the problem is saying that YM is 2 miles along YZ, and MX is 3 miles from M to X, which is the height of the triangle. So, if YM is 2 miles, then MZ would be 3 miles because YZ is 5 miles. But that contradicts M being the midpoint. So, maybe M is not the midpoint in terms of distance but in terms of the line segment.Wait, the problem says "midpoint M of YZ." So, M should be exactly halfway between Y and Z, making YM equal to MZ. If YZ is 5 miles, then YM and MZ should each be 2.5 miles. But the problem says YM is 2 miles and MX is 3 miles. This is conflicting.Maybe I'm overcomplicating this. Let's try to draw it out mentally. We have triangle XYZ with YZ as the base, 5 miles long. M is the midpoint, so YM = MZ = 2.5 miles. Elm Street goes from X to M, and they give us YM = 2 miles and MX = 3 miles. Wait, that doesn't fit because if M is the midpoint, YM should be 2.5 miles.Perhaps the problem is that YM is 2 miles along YZ, and MX is 3 miles from M to X, which is the height. So, the base YZ is 5 miles, and the height from X to YZ is 3 miles. Then, the area of triangle XYZ would be (1/2)*5*3 = 7.5 square miles. But we need the area of triangle XMY.Since M is the midpoint, triangle XMY would have half the base of XYZ, so YM = 2.5 miles, but the problem says YM is 2 miles. Wait, no, if M is the midpoint, YM should be 2.5 miles. But the problem says YM is 2 miles. Maybe the problem is misworded or I'm misinterpreting.Alternatively, maybe YM is 2 miles from Y to M along YZ, and MX is 3 miles from M to X, which is the height. So, the base YZ is 5 miles, and the height is 3 miles. Then, the area of XYZ is (1/2)*5*3 = 7.5. The area of XMY would be half of that because M is the midpoint, so 7.5/2 = 3.75. But that's not an option.Wait, the options are 2, 3, 4, 5, 6. So, 3.75 isn't there. Maybe I'm still misunderstanding.Let me try another approach. If M is the midpoint of YZ, then YM = MZ = 2.5 miles. But the problem says YM = 2 miles and MX = 3 miles. Maybe MX is the length of Elm Street, which is from X to M, and that's 3 miles. So, the height of the triangle from X to YZ is 3 miles. Then, the area of XYZ is (1/2)*5*3 = 7.5. The area of XMY would be half of that because M is the midpoint, so 7.5/2 = 3.75. Again, not an option.Wait, maybe the problem is that YM is 2 miles along YZ, and MX is 3 miles from M to X, which is the height. So, the base YZ is 5 miles, and the height is 3 miles. Then, the area of XYZ is 7.5. The area of XMY would be (1/2)*2*3 = 3 square miles. That's option B.Wait, but if M is the midpoint, YM should be 2.5 miles, not 2. Maybe the problem is that M is not the midpoint in terms of distance but in terms of the line segment. So, if YZ is 5 miles, and M is the midpoint, then YM = MZ = 2.5 miles. But the problem says YM = 2 miles and MX = 3 miles. So, maybe the problem is that M is not the midpoint of YZ in terms of distance but in terms of the line segment, meaning that YM = MZ in length, but the total YZ is 5 miles, so YM = MZ = 2.5 miles. But the problem says YM = 2 miles, which contradicts that.I'm getting confused. Maybe I should just go with the given numbers. If YM = 2 miles and MX = 3 miles, and YZ = 5 miles, then the area of XMY would be (1/2)*2*3 = 3 square miles. That's option B. Maybe the problem is that M is not exactly the midpoint in terms of distance, but just the midpoint of the line segment, so YM = MZ in terms of the line, not the actual distance. But that doesn't make sense because distance is a measure of length.Alternatively, maybe the problem is that M is the midpoint of YZ in terms of the line segment, so YM = MZ = 2.5 miles, but the problem gives YM = 2 miles and MX = 3 miles. Maybe the problem is misworded, and M is not the midpoint but some other point. But the problem clearly says "midpoint M of YZ."Wait, maybe the problem is that YM is 2 miles along YZ, and MX is 3 miles from M to X, which is the height. So, the base YZ is 5 miles, and the height is 3 miles. Then, the area of XYZ is 7.5, and the area of XMY is (1/2)*2*3 = 3. That makes sense, and 3 is an option.I think that's the way to go. So, the area of triangle XMY is 3 square miles.

💡First, I identify the side length of the initial equilateral triangle ( T_1 ), which is given as 50 cm.Next, I recognize that each subsequent triangle is formed by joining the midpoints of the previous triangle's sides. This means the side length of each new triangle is half the side length of the previous one.I calculate the side length of the second triangle ( T_2 ) by dividing the side length of ( T_1 ) by 2:[text{Side length of } T_2 = frac{50}{2} = 25 text{ cm}]To verify, I consider the sum of the perimeters of all triangles. The perimeter of each triangle is three times its side length. The perimeters form a geometric series:[text{Sum of perimeters} = 3 times 50 + 3 times 25 + 3 times 12.5 + ldots]Using the formula for the sum of an infinite geometric series ( S = frac{a}{1 - r} ), where ( a = 150 ) cm (the first term) and ( r = frac{1}{2} ) (the common ratio), I calculate:[S = frac{150}{1 - frac{1}{2}} = 300 text{ cm}]This confirms that the sum of the perimeters is indeed 300 cm, validating the side length of the second triangle as 25 cm.

💡Okay, so I'm trying to figure out which of these statements about triangle ABC are correct. Let me go through each one step by step.Starting with Statement A: If sin 2A = sin 2B, then triangle ABC is a right triangle. Hmm, I remember that sin θ = sin (π - θ), so if sin 2A = sin 2B, that could mean either 2A = 2B or 2A = π - 2B. Simplifying, that gives A = B or A + B = π/2. If A = B, then the triangle is isosceles, not necessarily right-angled. If A + B = π/2, then angle C would be π/2, making it a right triangle. But since it's not guaranteed that A + B = π/2, the triangle could be either isosceles or right-angled. So Statement A isn't necessarily true. I think A is incorrect.Moving on to Statement B: If sin A > sin B, then A > B. I recall that in a triangle, the larger angle is opposite the longer side. Also, from the Law of Sines, a/sin A = b/sin B = 2R, where R is the circumradius. So if sin A > sin B, then a > b, which implies A > B. That makes sense. So Statement B seems correct.Now, Statement C: If a=12, b=10, B=60°, then the number of triangles ABC that satisfy the conditions is two. Let me use the Law of Sines here. The Law of Sines says a/sin A = b/sin B. Plugging in the values, 12/sin A = 10/sin 60°. Sin 60° is √3/2, so 10/(√3/2) = 20/√3 ≈ 11.547. So 12/sin A ≈ 11.547, which means sin A ≈ 12 / 11.547 ≈ 1.039. Wait, that's greater than 1, which isn't possible because the sine of an angle can't exceed 1. So does that mean there's no such triangle? But the question says the number is two. Hmm, maybe I made a mistake. Let me check again. If a=12, b=10, and B=60°, then using the Law of Sines, sin A = (a * sin B)/b = (12 * sin 60°)/10 = (12 * √3/2)/10 = (6√3)/10 ≈ 1.039. Yeah, that's still over 1, which isn't possible. So actually, there are no such triangles. But the statement says two, so I think Statement C is incorrect.Finally, Statement D: If sin²A + sin²B < sin²C, then triangle ABC is an obtuse triangle. Let me think about this. Using the Law of Sines, sin A = a/(2R), sin B = b/(2R), sin C = c/(2R). So sin²A + sin²B < sin²C translates to (a² + b²)/(4R²) < c²/(4R²), which simplifies to a² + b² < c². From the Law of Cosines, c² = a² + b² - 2ab cos C. So if a² + b² < c², then substituting, we get a² + b² < a² + b² - 2ab cos C, which simplifies to 0 < -2ab cos C. Since a and b are positive, this implies that cos C < 0, meaning angle C is greater than 90°, so the triangle is obtuse. Therefore, Statement D is correct.So, summarizing my thoughts: Statement A is incorrect, Statement B is correct, Statement C is incorrect, and Statement D is correct. So the correct statements are B and D.**Final Answer**The correct statements are boxed{B} and boxed{D}.boxed{BD}