answer：First, I need to determine the probability that both A and B tell the truth at the same time.I know that A tells the truth 55% of the time, which is 0.55 in decimal form.Similarly, B tells the truth 60% of the time, which is 0.60 in decimal form.Since the events are independent, I can find the joint probability by multiplying the individual probabilities.So, 0.55 multiplied by 0.60 equals 0.33.Therefore, the probability that both A and B tell the truth at the same time is 0.33, or 33%.

answer：First, I need to determine the unknown dimension of the tank, which I'll call ( x ).The tank has three dimensions: ( x ) feet, 5 feet, and 2 feet. To find the total surface area that needs insulation, I'll calculate the area of each of the six sides of the rectangular tank.1. **Two sides** with dimensions ( x ) by 5 feet: [ 2 times (x times 5) = 10x text{ square feet} ]2. **Two sides** with dimensions 5 feet by 2 feet: [ 2 times (5 times 2) = 20 text{ square feet} ]3. **Two sides** with dimensions ( x ) by 2 feet: [ 2 times (x times 2) = 4x text{ square feet} ]Adding these together gives the total surface area:[10x + 20 + 4x = 14x + 20 text{ square feet}]Each square foot of insulation costs 20, so the total cost to cover the tank is:[(14x + 20) times 20 text{ dollars}]We know the total cost is 1240, so I'll set up the equation:[(14x + 20) times 20 = 1240]To solve for ( x ), I'll divide both sides by 20:[14x + 20 = 62]Next, I'll subtract 20 from both sides:[14x = 42]Finally, I'll divide both sides by 14:[x = 3]So, the length of the first dimension of the tank is 3 feet.

answer：First, identify the total number of cans, which is 24.Out of these, 10 cans hold 8 liters each. To find the total oil in these 10 cans, multiply the number of cans by the oil per can:10 cans × 8 liters/can = 80 liters.The remaining cans are 24 - 10 = 14 cans, and each of these holds 15 liters. Calculate the total oil in these 14 cans:14 cans × 15 liters/can = 210 liters.Finally, add the oil from both groups of cans to find the total oil:80 liters + 210 liters = 290 liters.

answer：First, I note that there were initially 30 people in line for the bumper cars.Some people got tired of waiting and left the line. Let's denote the number of people who left as ( L ).After these ( L ) people left, 5 more people joined the line. This means the total number of people in line became ( 30 - L + 5 ).According to the problem, the final number of people in line is 25. Therefore, I can set up the equation:[30 - L + 5 = 25]Combining like terms:[35 - L = 25]To solve for ( L ), I subtract 25 from both sides:[35 - 25 = L][10 = L]Thus, 10 people left the line.

answer：Okay, so I have this problem about marbles in a basket. There are white, orange, and purple marbles. The problem says:"All but 10 are white, all but 12 are orange, and all but 5 are purple. How many marbles are in the basket?"Hmm, let me try to understand what this means. When it says "all but 10 are white," that should mean if I take away the white marbles, there are 10 left. So, the number of orange and purple marbles together is 10.Similarly, "all but 12 are orange" means if I take away the orange marbles, there are 12 left. So, the number of white and purple marbles together is 12.And "all but 5 are purple" means if I take away the purple marbles, there are 5 left. So, the number of white and orange marbles together is 5.Let me write these down as equations. Let me denote:- W = number of white marbles- O = number of orange marbles- P = number of purple marblesFrom the problem:1. All but 10 are white: So, O + P = 102. All but 12 are orange: So, W + P = 123. All but 5 are purple: So, W + O = 5So now I have three equations:1. O + P = 102. W + P = 123. W + O = 5I need to find the total number of marbles, which is W + O + P.Hmm, okay, so I have three equations with three variables. Maybe I can solve for each variable step by step.Let me see. From equation 3: W + O = 5. Maybe I can express W in terms of O. So, W = 5 - O.Then, I can substitute this into equation 2: W + P = 12. So, substituting W:(5 - O) + P = 12Simplify that:5 - O + P = 12Let me rearrange:-P + O = 5 - 12Wait, that would be -P + O = -7. Hmm, maybe I should rearrange differently.Wait, 5 - O + P = 12Subtract 5 from both sides:-O + P = 12 - 5-O + P = 7So, P = O + 7Okay, so P is equal to O plus 7.Now, let's go back to equation 1: O + P = 10But since P = O + 7, substitute that into equation 1:O + (O + 7) = 10Simplify:2O + 7 = 10Subtract 7 from both sides:2O = 3Divide both sides by 2:O = 1.5Wait, O is 1.5? That seems odd because marbles are whole objects, right? So, 1.5 marbles? Hmm, maybe the problem allows for fractional marbles? Or perhaps I made a mistake.Let me check my steps again.From equation 3: W + O = 5, so W = 5 - O.Substitute into equation 2: W + P = 12:(5 - O) + P = 12So, 5 - O + P = 12Subtract 5: -O + P = 7So, P = O + 7Then, substitute into equation 1: O + P = 10O + (O + 7) = 102O + 7 = 102O = 3O = 1.5Hmm, same result. So, O is 1.5. Maybe the problem is designed this way, allowing for fractional marbles? Or perhaps I misinterpreted the problem.Wait, let me read the problem again."All but 10 are white, all but 12 are orange, and all but 5 are purple."So, all but 10 are white: So, the number of non-white marbles is 10. That is, orange + purple = 10.Similarly, all but 12 are orange: So, non-orange marbles are 12, which is white + purple = 12.All but 5 are purple: So, non-purple marbles are 5, which is white + orange = 5.So, my equations are correct.So, solving them gives O = 1.5, which is 3/2.Then, P = O + 7 = 1.5 + 7 = 8.5And W = 5 - O = 5 - 1.5 = 3.5So, W = 3.5, O = 1.5, P = 8.5Total marbles: 3.5 + 1.5 + 8.5 = 13.5Hmm, 13.5 marbles. That's 13 and a half marbles. So, the answer is 13.5.Looking at the options:(A) 12(B) 13.5(C) 14(D) 15(E) 15.5So, 13.5 is option B.But wait, marbles are physical objects, so having half a marble seems odd. Maybe the problem is designed to allow for fractional marbles, or perhaps I made a mistake.Wait, let me double-check.If O = 1.5, P = 8.5, W = 3.5Then, O + P = 1.5 + 8.5 = 10, which matches the first condition.W + P = 3.5 + 8.5 = 12, which matches the second condition.W + O = 3.5 + 1.5 = 5, which matches the third condition.So, all conditions are satisfied.Therefore, even though it seems odd, the total number of marbles is 13.5.So, the answer is B) 13.5.I think that's correct. Maybe the problem allows for fractional marbles, or perhaps it's a trick question. But mathematically, it works out to 13.5.

answer：Okay, so I have this problem here about a circle with a radius of 3 cm, and I need to find the length of the longest chord in the circle. The options are A: 3 cm, B: 6 cm, C: 1.5 cm, and D: √3 cm. Hmm, let me think about this step by step.First, I remember that a chord is a straight line connecting two points on the circumference of a circle. The longest possible chord in a circle is actually the diameter, right? Because the diameter passes through the center of the circle and connects two opposite points on the circumference. So, if I can find the diameter of this circle, that should give me the length of the longest chord.Given that the radius is 3 cm, I recall that the diameter is twice the radius. So, if the radius is 3 cm, then the diameter should be 2 times 3 cm, which is 6 cm. That seems straightforward.Wait, let me make sure I'm not missing anything here. Is there any other chord that could be longer than the diameter? I don't think so because the diameter is the maximum distance between any two points on the circle. Any other chord would be shorter than or equal to the diameter. So, in this case, the diameter is definitely the longest chord.Looking back at the options, B is 6 cm, which matches the diameter I calculated. So, that should be the correct answer.Just to double-check, let me visualize the circle. If I draw a circle with radius 3 cm, the diameter would stretch from one end of the circle to the other, passing through the center. That distance would be twice the radius, so 6 cm. Yeah, that makes sense.I don't think any of the other options make sense. Option A is 3 cm, which is just the radius, not the diameter. Option C is 1.5 cm, which is even smaller than the radius, so that's definitely not the longest chord. Option D is √3 cm, which is approximately 1.732 cm, still much smaller than the radius. So, none of these other options are longer than the diameter.Therefore, I'm confident that the longest chord in the circle is 6 cm, which is option B.