💡Alright, so I have this problem about an isosceles triangle with one side of length 3 and another side of length 6. I need to find its perimeter. Hmm, okay, let's break this down step by step.First, I remember that an isosceles triangle has at least two sides that are equal in length. So, in this case, we have two sides given: 3 and 6. Now, I need to figure out which sides are equal.There are two possibilities here:1. The two equal sides are both 3 units long, and the third side is 6 units.2. The two equal sides are both 6 units long, and the third side is 3 units.Let me consider the first possibility: if the two equal sides are 3 units each, then the sides of the triangle would be 3, 3, and 6. But wait a minute, does this satisfy the triangle inequality theorem? The theorem states that the sum of the lengths of any two sides must be greater than the length of the remaining side.So, let's check:- 3 + 3 = 6, which is equal to the third side, not greater. That means this doesn't satisfy the triangle inequality theorem. Therefore, a triangle with sides 3, 3, and 6 is not possible.Okay, so the first possibility is out. Let's move on to the second possibility: the two equal sides are both 6 units long, and the third side is 3 units. So, the sides would be 6, 6, and 3.Again, I need to check the triangle inequality theorem:- 6 + 6 = 12, which is greater than 3.- 6 + 3 = 9, which is greater than 6.- 6 + 3 = 9, which is greater than 6.All these satisfy the triangle inequality theorem, so this is a valid triangle.Now, to find the perimeter, I just need to add up all the sides:6 + 6 + 3 = 15.So, the perimeter of the triangle is 15 units.Wait, but I see that one of the answer choices is "15 or 18." Why is there another option? Did I miss something?Let me think again. Maybe I should consider if there's another way to arrange the sides. But in an isosceles triangle, only two sides are equal, so the only other possibility would be if the two equal sides are 3, but we saw that doesn't work. So, I don't think there's another valid arrangement.Hmm, maybe the question is trying to trick me by suggesting that both 15 and 18 could be possible perimeters. But based on my calculations, only 15 is valid. So, I think the correct answer is 15.But just to be thorough, let's imagine if the triangle were equilateral, meaning all sides are equal. But in that case, all sides would have to be either 3 or 6, which isn't the case here since we have both 3 and 6. So, that's not applicable.Another thought: could the triangle have sides 3, 6, and 6, making the perimeter 15, or sides 3, 3, and 6, which we already determined isn't possible? So, only 15 is valid.Therefore, I'm confident that the perimeter is 15 units.

💡First, I need to understand the problem: finding how many three-digit numbers exist where the units digit is at least three times the tens digit.I'll start by defining the digits of a three-digit number. Let the hundreds digit be ( h ), the tens digit be ( t ), and the units digit be ( u ). Since it's a three-digit number, ( h ) can range from 1 to 9, ( t ) from 0 to 9, and ( u ) from 0 to 9.The key condition is ( u geq 3t ). I'll analyze each possible value of ( t ) to determine the valid values of ( u ) that satisfy this condition.For ( t = 0 ):- ( u ) can be any digit from 0 to 9, giving 10 possibilities.For ( t = 1 ):- ( u ) must be at least 3, so ( u ) can be 3 to 9, giving 7 possibilities.For ( t = 2 ):- ( u ) must be at least 6, so ( u ) can be 6 to 9, giving 4 possibilities.For ( t = 3 ):- ( u ) must be at least 9, so ( u ) can only be 9, giving 1 possibility.For ( t = 4 ) to ( t = 9 ):- ( 3t ) exceeds the maximum value of ( u ) (which is 9), so there are no valid ( u ) values for these ( t ) values.Next, I'll calculate the total number of valid combinations for ( t ) and ( u ):- ( t = 0 ): 10- ( t = 1 ): 7- ( t = 2 ): 4- ( t = 3 ): 1- ( t = 4 ) to ( t = 9 ): 0Adding these up: ( 10 + 7 + 4 + 1 = 22 ).Since the hundreds digit ( h ) can be any digit from 1 to 9, there are 9 possible choices for ( h ).Finally, the total number of three-digit numbers that satisfy the condition is:[22 times 9 = 198]

💡Okay, so I have this problem here: "A triangle has integer side lengths, and the length of the longest side is 11. How many such triangles are there?" The options are A. 32, B. 34, C. 36, D. 40. Hmm, I need to figure out how many triangles satisfy these conditions.First, I remember that for a set of three sides to form a triangle, they must satisfy the triangle inequality theorem. That is, the sum of the lengths of any two sides must be greater than the length of the remaining side. Since the longest side is 11, let's denote the sides as a, b, and 11, where a ≤ b ≤ 11.So, the triangle inequalities we need to satisfy are:1. a + b > 112. a + 11 > b3. b + 11 > aBut since a ≤ b ≤ 11, some of these inequalities might automatically hold. For example, the third inequality, b + 11 > a, will always be true because b is at least as big as a, and 11 is positive. Similarly, the second inequality, a + 11 > b, simplifies to a > b - 11. But since a ≤ b, and b ≤ 11, b - 11 is less than or equal to 0, so a > something less than or equal to 0. Since a is a positive integer, this inequality is automatically satisfied as well.So, the only inequality we really need to worry about is the first one: a + b > 11. That's the key condition here.Now, since a and b are integers and a ≤ b ≤ 11, I can think of this as a counting problem where I need to count all pairs (a, b) such that a ≤ b ≤ 11 and a + b > 11.Let me structure this systematically. I can fix the value of a and then determine the possible values of b for each a.Starting with a = 1:- Since a = 1, b must satisfy 1 ≤ b ≤ 11 and 1 + b > 11, which simplifies to b > 10. So, b can only be 11. That's 1 value.a = 2:- 2 ≤ b ≤ 11 and 2 + b > 11 ⇒ b > 9. So, b can be 10 or 11. That's 2 values.a = 3:- 3 ≤ b ≤ 11 and 3 + b > 11 ⇒ b > 8. So, b can be 9, 10, or 11. That's 3 values.a = 4:- 4 ≤ b ≤ 11 and 4 + b > 11 ⇒ b > 7. So, b can be 8, 9, 10, or 11. That's 4 values.a = 5:- 5 ≤ b ≤ 11 and 5 + b > 11 ⇒ b > 6. So, b can be 7, 8, 9, 10, or 11. That's 5 values.a = 6:- 6 ≤ b ≤ 11 and 6 + b > 11 ⇒ b > 5. Since b is at least 6, all b from 6 to 11 satisfy this. So, b can be 6, 7, 8, 9, 10, or 11. That's 6 values.a = 7:- 7 ≤ b ≤ 11 and 7 + b > 11 ⇒ b > 4. But since b is at least 7, all b from 7 to 11 satisfy this. So, b can be 7, 8, 9, 10, or 11. That's 5 values.Wait, hold on. When a = 7, b starts at 7, and 7 + 7 = 14, which is greater than 11, so all b from 7 to 11 are valid. So, 5 values.Similarly, for a = 8:- 8 ≤ b ≤ 11 and 8 + b > 11 ⇒ b > 3. But since b is at least 8, all b from 8 to 11 are valid. So, 4 values.a = 9:- 9 ≤ b ≤ 11 and 9 + b > 11 ⇒ b > 2. Since b is at least 9, all b from 9 to 11 are valid. So, 3 values.a = 10:- 10 ≤ b ≤ 11 and 10 + b > 11 ⇒ b > 1. Since b is at least 10, all b from 10 to 11 are valid. So, 2 values.a = 11:- 11 ≤ b ≤ 11 and 11 + b > 11 ⇒ b > 0. Since b is exactly 11, this is valid. So, 1 value.Now, let me list out the number of valid b's for each a:- a=1: 1- a=2: 2- a=3: 3- a=4: 4- a=5: 5- a=6: 6- a=7: 5- a=8: 4- a=9: 3- a=10: 2- a=11: 1Now, let's add these up:1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1Let me compute this step by step:1 + 2 = 33 + 3 = 66 + 4 = 1010 + 5 = 1515 + 6 = 2121 + 5 = 2626 + 4 = 3030 + 3 = 3333 + 2 = 3535 + 1 = 36So, the total number of such triangles is 36.Wait, but let me double-check my addition to make sure I didn't make a mistake:1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1Group them as follows:(1 + 2 + 3 + 4 + 5) + (6 + 5 + 4 + 3 + 2 + 1)Compute each group:First group: 1+2=3, 3+3=6, 6+4=10, 10+5=15Second group: 6+5=11, 11+4=15, 15+3=18, 18+2=20, 20+1=21Now, add the two groups: 15 + 21 = 36Yes, that's correct.Alternatively, I can think of this as a symmetric pattern. The counts increase up to a=6 and then decrease symmetrically. So, it's like 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1. The sum of this symmetric sequence is 2*(1+2+3+4+5) + 6 = 2*15 +6=36.Yes, that also gives 36.Therefore, the number of triangles is 36, which is option C.**Final Answer**boxed{C}

💡First, I need to determine the revolutions per minute (r.p.m.) of the bus wheel. To do this, I'll start by converting the bus's speed from kilometers per hour to centimeters per minute.The bus speed is 66 km/h. Since 1 kilometer equals 100,000 centimeters, I can convert the speed as follows:66 km/h = 66 × 100,000 cm/h = 6,600,000 cm/h.Next, I'll convert the speed from centimeters per hour to centimeters per minute by dividing by 60:6,600,000 cm/h ÷ 60 = 110,000 cm/min.Now, I'll calculate the circumference of the wheel using the formula:Circumference = 2 × π × radius.Given the radius of the wheel is 175 cm:Circumference = 2 × π × 175 cm ≈ 1,100 cm.Finally, to find the revolutions per minute, I'll divide the distance traveled per minute by the circumference of the wheel:Revolutions per minute = 110,000 cm/min ÷ 1,100 cm ≈ 100 r.p.m.

💡First, determine the relative speed between Samuel and Gabriella. Since they are moving in the same direction, subtract Gabriella's speed from Samuel's speed:[text{Relative speed} = 30 text{ mph} - 22 text{ mph} = 8 text{ mph}]Next, calculate the time it takes for Samuel to cover the initial 1 mile distance to overtake Gabriella:[text{Time to overtake} = frac{1 text{ mile}}{8 text{ mph}} = frac{1}{8} text{ hour}]After overtaking, Samuel continues to move ahead of Gabriella. Calculate the time until Gabriella is 1 mile behind Samuel:[text{Time until 1 mile behind} = frac{1 text{ mile}}{8 text{ mph}} = frac{1}{8} text{ hour}]Add both times to find the total observation time:[text{Total time} = frac{1}{8} text{ hour} + frac{1}{8} text{ hour} = frac{1}{4} text{ hour}]Convert the total time from hours to minutes:[frac{1}{4} text{ hour} times 60 text{ minutes/hour} = 15 text{ minutes}]

💡Okay, so I have this problem here: Given the imaginary unit ( i ), find the complex number ( z ) that satisfies ( frac{z}{z - i} = i ). Hmm, complex numbers can be a bit tricky, but let me try to work through this step by step.First, I need to understand what the equation is saying. It's saying that when you divide ( z ) by ( z - i ), you get ( i ). So, ( z ) divided by ( z - i ) equals ( i ). That means ( z ) is some multiple of ( i ) when compared to ( z - i ). I think the best way to approach this is to solve for ( z ) algebraically.Let me write down the equation again:[frac{z}{z - i} = i]To get rid of the fraction, I can multiply both sides of the equation by ( z - i ). That should help me isolate ( z ). So, multiplying both sides by ( z - i ):[z = i(z - i)]Alright, now I have ( z ) on both sides, but at least the fraction is gone. Let me expand the right-hand side. Distribute the ( i ):[z = iz - i^2]Wait, ( i^2 ) is equal to ( -1 ), right? So, substituting that in:[z = iz - (-1) implies z = iz + 1]Hmm, okay, so now I have ( z = iz + 1 ). I need to get all the terms involving ( z ) on one side. Let me subtract ( iz ) from both sides to move it to the left:[z - iz = 1]Now, I can factor out ( z ) from the left-hand side:[z(1 - i) = 1]Great, so now I have ( z ) multiplied by ( (1 - i) ) equals 1. To solve for ( z ), I need to divide both sides by ( (1 - i) ):[z = frac{1}{1 - i}]But wait, having a complex number in the denominator isn't usually preferred. I remember that to simplify expressions like this, we can multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of ( 1 - i ) is ( 1 + i ). Let me do that:[z = frac{1 times (1 + i)}{(1 - i)(1 + i)}]Now, let's simplify the denominator. When I multiply ( (1 - i)(1 + i) ), it's a difference of squares, so:[(1 - i)(1 + i) = 1^2 - (i)^2 = 1 - (-1) = 1 + 1 = 2]So, the denominator simplifies to 2. The numerator is ( 1 + i ). Therefore, the expression becomes:[z = frac{1 + i}{2}]Which can also be written as:[z = frac{1}{2} + frac{1}{2}i]Let me double-check my steps to make sure I didn't make any mistakes. Starting from the original equation, I multiplied both sides by ( z - i ), expanded, substituted ( i^2 = -1 ), moved terms around, factored out ( z ), and then divided. Then I multiplied numerator and denominator by the conjugate to simplify. Each step seems logical and follows algebraic rules. Just to be thorough, let me plug this value of ( z ) back into the original equation to verify if it holds true. Let ( z = frac{1}{2} + frac{1}{2}i ). Then ( z - i = frac{1}{2} + frac{1}{2}i - i = frac{1}{2} - frac{1}{2}i ).Now, compute ( frac{z}{z - i} ):[frac{frac{1}{2} + frac{1}{2}i}{frac{1}{2} - frac{1}{2}i}]To simplify this, I can multiply numerator and denominator by 2 to eliminate the fractions:[frac{1 + i}{1 - i}]Again, to simplify, multiply numerator and denominator by the conjugate of the denominator, which is ( 1 + i ):[frac{(1 + i)(1 + i)}{(1 - i)(1 + i)} = frac{(1 + 2i + i^2)}{1 - i^2}]Simplify numerator and denominator:Numerator: ( 1 + 2i + (-1) = 2i )Denominator: ( 1 - (-1) = 2 )So, the expression becomes:[frac{2i}{2} = i]Which matches the original equation. Therefore, my solution is correct.So, the complex number ( z ) that satisfies the equation ( frac{z}{z - i} = i ) is ( frac{1}{2} + frac{1}{2}i ).