💡First, determine the cost of one T-shirt. Since each T-shirt costs 6 more than a pair of socks, which are 4, the cost of one T-shirt is:[4 + 6 = 10 text{ dollars}]Next, calculate the cost of one cap. Each cap costs 3 less than a T-shirt, so the cost of one cap is:[10 - 3 = 7 text{ dollars}]Now, calculate the total cost for one member. Each member requires one pair of socks, one T-shirt, and one cap for both home and away games, so the total cost for one member is:[2 times (4 + 10 + 7) = 2 times 21 = 42 text{ dollars}]Finally, determine the number of members in the league by dividing the total expenditure by the cost per member:[frac{3144}{42} = 74.8571 approx 75 text{ (rounding to the nearest whole number)}]Thus, the number of members in the Compton Basketball League is (boxed{mathrm{(B)} 75}).

💡First, I need to determine the total number of animals on the street. I know that two-thirds of the animals are cats and one-third are dogs.Let’s denote the total number of animals as ( A ).Since one-third of the animals are dogs, the number of dogs ( D ) can be expressed as:[ D = frac{A}{3} ]Each dog has 4 legs, so the total number of legs for all the dogs is:[ 4D ]According to the problem, the total number of legs of dogs is 400:[ 4D = 400 ]Substituting ( D ) with ( frac{A}{3} ):[ 4left(frac{A}{3}right) = 400 ]Simplifying the equation:[ frac{4A}{3} = 400 ]To solve for ( A ), multiply both sides by 3:[ 4A = 1200 ]Finally, divide both sides by 4:[ A = 300 ]Therefore, there are 300 animals on the street in total.

💡First, calculate the total number of popsicle sticks brought by the boys by multiplying the number of boys by the number of popsicle sticks each boy brought.Then, calculate the total number of popsicle sticks brought by the girls by multiplying the number of girls by the number of popsicle sticks each girl brought.Finally, subtract the total number of popsicle sticks brought by the girls from the total number brought by the boys to find out how many fewer popsicle sticks the girls brought.

💡First, I recognize that in a geometric sequence, each term is obtained by multiplying the previous term by the common ratio ( q ).Given that ( a_7 = 10 ) and the common ratio ( q = -2 ), I need to find ( a_{10} ).To find ( a_{10} ), I can use the relationship between consecutive terms in a geometric sequence. Specifically, ( a_{10} ) can be expressed in terms of ( a_7 ) and the common ratio ( q ).The formula to find ( a_{10} ) is:[a_{10} = a_7 times q^{(10 - 7)} = 10 times (-2)^3]Calculating ( (-2)^3 ):[(-2)^3 = -8]Now, multiply this result by 10:[10 times (-8) = -80]Therefore, ( a_{10} = -80 ).

💡Okay, so I have this problem here: "The converse of the proposition 'If two angles are equal, then their complements are equal' is." Alright, I need to figure out what the converse of this statement is. Hmm, let me start by recalling what a converse is in logic.From what I remember, a conditional statement has the form "If P, then Q." The converse of this statement would be "If Q, then P." So, basically, you switch the hypothesis and the conclusion. Got it. So, in this case, the original statement is "If two angles are equal, then their complements are equal." Let me break this down. The original statement is saying that if we have two angles that are equal in measure, then their complements will also be equal. So, if angle A equals angle B, then the complement of angle A equals the complement of angle B. That makes sense because if two angles are equal, subtracting each from 90 degrees (since complements add up to 90) would still give equal results.Now, to find the converse, I need to switch the hypothesis and the conclusion. So, instead of starting with "If two angles are equal," I should start with "If their complements are equal." Then, the conclusion would be "then the two angles are equal." So, putting that together, the converse should be: "If the complements of two angles are equal, then the two angles are equal."Wait, let me make sure I'm doing this right. The original statement is P → Q, where P is "two angles are equal" and Q is "their complements are equal." So, the converse would be Q → P, which is "If their complements are equal, then the two angles are equal." Yeah, that seems right.But hold on, is this converse always true? I mean, in the original statement, if two angles are equal, their complements are definitely equal because you're just subtracting the same measure from 90 degrees. But does the converse hold? If two angles have equal complements, does that necessarily mean the angles themselves are equal?Let me think about it. Suppose angle A and angle B have complements that are equal. That means 90° - A = 90° - B. If I solve for A and B, subtracting 90° from both sides gives -A = -B, and multiplying both sides by -1 gives A = B. So, yes, if the complements are equal, the angles must be equal. So, the converse is actually true in this case.But wait, is that always the case? What if the angles weren't in the same plane or something? No, angles are just measures, so regardless of their position, if their complements are equal, the angles themselves must be equal. Okay, so the converse is not only the logical switch but also a true statement.Let me try to rephrase the converse to make sure I understand it correctly. The original statement was about equality leading to equal complements, and the converse is about equal complements leading to equality. So, if I have two angles, and I know that their complements are the same, then I can conclude that the angles themselves are the same. That seems logical.Is there a way to test this? Maybe with some numbers. Let's say angle A is 30°, so its complement is 60°. If angle B is also 30°, its complement is 60°, so their complements are equal, and the angles are equal. Now, what if angle A is 30° and angle B is 40°? The complement of A is 60°, and the complement of B is 50°, which are not equal. So, unequal angles have unequal complements. Conversely, if I have two angles with complements of 60°, then both angles must be 30°, right? Because 90° - 60° = 30°. So, if two angles have the same complement, they must be the same angle. That checks out.I think I've got it. The converse is formed by switching the hypothesis and conclusion, and in this case, it's a valid and true statement. So, the converse of "If two angles are equal, then their complements are equal" is "If the complements of two angles are equal, then the two angles are equal."Just to make sure I didn't make any mistakes, let me think about another example. Suppose angle X is 45°, so its complement is 45°. If angle Y is also 45°, its complement is 45°, so their complements are equal, and the angles are equal. Now, if angle X is 50°, its complement is 40°, and if angle Y is 50°, its complement is 40°, so again, equal complements mean equal angles. What if angle X is 20°, complement is 70°, and angle Y is 70°, complement is 20°. Wait, hold on, that's different. If angle X is 20°, complement is 70°, and angle Y is 70°, complement is 20°. So, their complements are different. So, unequal angles have unequal complements. So, if their complements are equal, the angles must be equal.Wait, but in this case, if angle X is 20°, complement is 70°, and angle Y is 70°, complement is 20°, which are different. So, if two angles have equal complements, then they must be equal. So, if angle X's complement equals angle Y's complement, then angle X equals angle Y. Yeah, that seems consistent.So, I think I've confirmed that the converse is correct. It's just a matter of switching the hypothesis and conclusion, and in this case, it still holds true. So, the converse is "If the complements of two angles are equal, then the two angles are equal."I guess another way to think about it is in terms of functions. The complement of an angle is a function that maps an angle to another angle such that their sum is 90°. So, if two angles have the same complement, then they must be the same angle because the function is one-to-one. That is, each angle has a unique complement, and each complement corresponds to exactly one angle. So, if two angles have the same complement, they must be identical.Yeah, that makes sense. So, the converse is not only a logical switch but also maintains the truth value because the function is invertible. Therefore, the converse is indeed a true statement.I think I've thought this through enough. The converse is formed by switching the parts of the original statement, and in this case, it's a valid and true statement. So, I'm confident that the converse is "If the complements of two angles are equal, then the two angles are equal."

💡First, I need to calculate the dot product of vectors **a** and **b**. The dot product is found by multiplying the corresponding components of the vectors and then summing the results.Given:- Vector **a** has components (5, 3)- Vector **b** has components (4, 2)I will multiply the x-components and the y-components separately:- 5 multiplied by 4 equals 20- 3 multiplied by 2 equals 6Next, I will add these products together:- 20 plus 6 equals 26Therefore, the dot product of vectors **a** and **b** is 26.