💡First, I need to determine Alec's remaining budget after paying the entrance fee. His total budget is 200, and the entrance fee is 5, so he has 195 left for purchasing the shirts.Next, I'll consider the discount Alec receives for buying more than 15 shirts. Since he is purchasing 20 shirts, he qualifies for a 10% discount on the total price before tax. Let ( p ) represent the price per shirt before any discounts or taxes. The total cost for 20 shirts before the discount is ( 20p ).Applying the 10% discount, the total cost becomes:[20p times 0.9 = 18p]After the discount, a 5% sales tax is added to the discounted total. The total cost including tax is:[18p times 1.05 = 18.9p]Now, I'll set up the inequality to ensure the total cost does not exceed Alec's remaining budget:[18.9p leq 195]To find the maximum price per shirt, I'll solve for ( p ):[p leq frac{195}{18.9} approx 10.31746]Since the price per shirt must be a whole number, the greatest possible price per shirt Alec can afford is 10.

💡Okay, so I need to find the maximum value of the function ( f(x) = x^3 - 3x + 1 ) on the closed interval ([-3, 0]). Hmm, let me think about how to approach this. I remember from my calculus class that to find the maximum or minimum values of a function on a closed interval, I need to check both the critical points inside the interval and the endpoints of the interval. First, I should find the critical points of the function. Critical points occur where the derivative is zero or undefined. Since this function is a polynomial, its derivative will be defined everywhere, so I just need to find where the derivative equals zero.Let me compute the derivative of ( f(x) ). The derivative of ( x^3 ) is ( 3x^2 ), the derivative of ( -3x ) is ( -3 ), and the derivative of the constant term ( 1 ) is ( 0 ). So, putting it all together, the derivative ( f'(x) ) is:[ f'(x) = 3x^2 - 3 ]Now, I need to set this derivative equal to zero and solve for ( x ):[ 3x^2 - 3 = 0 ]Let me factor out a 3 to simplify:[ 3(x^2 - 1) = 0 ]So, ( x^2 - 1 = 0 ), which means:[ x^2 = 1 ]Taking the square root of both sides gives:[ x = pm 1 ]So, the critical points are at ( x = 1 ) and ( x = -1 ). Now, I need to determine which of these critical points lie within the interval ([-3, 0]). The critical point at ( x = 1 ) is outside this interval because 1 is greater than 0. However, the critical point at ( x = -1 ) is within the interval since -1 is between -3 and 0. Therefore, I only need to consider ( x = -1 ) as a critical point within the interval.Next, I need to evaluate the function ( f(x) ) at the critical point ( x = -1 ) and at the endpoints of the interval, which are ( x = -3 ) and ( x = 0 ). The maximum value will be the largest of these three values.Let me compute ( f(-3) ):[ f(-3) = (-3)^3 - 3(-3) + 1 ][ f(-3) = -27 + 9 + 1 ][ f(-3) = -17 ]Okay, that's pretty low. Now, let's compute ( f(-1) ):[ f(-1) = (-1)^3 - 3(-1) + 1 ][ f(-1) = -1 + 3 + 1 ][ f(-1) = 3 ]Hmm, that's a much higher value. Finally, let's compute ( f(0) ):[ f(0) = (0)^3 - 3(0) + 1 ][ f(0) = 0 - 0 + 1 ][ f(0) = 1 ]So, summarizing the values:- ( f(-3) = -17 )- ( f(-1) = 3 )- ( f(0) = 1 )Comparing these, the largest value is 3 at ( x = -1 ). Therefore, the maximum value of the function ( f(x) ) on the interval ([-3, 0]) is 3.Wait, just to make sure I didn't make any calculation errors, let me double-check the computations:For ( f(-3) ):- ( (-3)^3 = -27 )- ( -3(-3) = 9 )- Adding 1: ( -27 + 9 + 1 = -17 ). That seems correct.For ( f(-1) ):- ( (-1)^3 = -1 )- ( -3(-1) = 3 )- Adding 1: ( -1 + 3 + 1 = 3 ). That's correct too.For ( f(0) ):- ( 0^3 = 0 )- ( -3(0) = 0 )- Adding 1: ( 0 + 0 + 1 = 1 ). Yep, that's right.So, all the computations check out. Therefore, I'm confident that the maximum value is indeed 3 at ( x = -1 ).I also remember that sometimes, especially with cubic functions, the behavior at the endpoints can be tricky. But in this case, since the interval is from -3 to 0, and the function is a cubic, it tends to negative infinity as ( x ) approaches negative infinity, but within our interval, it's bounded. So, the function doesn't go to infinity here, which is why the maximum is at the critical point inside the interval.Just to visualize, if I think about the graph of ( f(x) = x^3 - 3x + 1 ), it's a cubic curve that has a local maximum at ( x = -1 ) and a local minimum at ( x = 1 ). Since we're only looking from -3 to 0, the peak at ( x = -1 ) is the highest point in that interval, and the function decreases towards both ends of the interval, reaching -17 at ( x = -3 ) and 1 at ( x = 0 ).So, putting it all together, the maximum value is 3.**Final Answer**The maximum value of the function ( f(x) = x^3 - 3x + 1 ) on the interval ([-3, 0]) is boxed{3}.

💡Okay, so I need to figure out the converse of the proposition "Vertical angles are equal." Hmm, let's start by understanding what "converse" means in this context. I remember from my math classes that the converse of a statement is formed by switching the hypothesis and the conclusion. So, if the original statement is "If A, then B," the converse would be "If B, then A."Alright, applying that to "Vertical angles are equal." First, I need to identify the hypothesis and the conclusion in this statement. The original statement seems to be saying that whenever two angles are vertical, they are equal. So, in logical terms, it's like "If two angles are vertical, then they are equal." That makes the hypothesis "two angles are vertical" and the conclusion "they are equal."Now, to find the converse, I need to switch these two parts. So, the converse would be "If two angles are equal, then they are vertical." That seems straightforward, but I want to make sure I'm not missing anything. Let me think about it a bit more.Are there any cases where two angles are equal but not vertical? I know that vertical angles are formed when two lines intersect, creating opposite angles that are equal. But there are other scenarios where angles can be equal without being vertical. For example, in an isosceles triangle, the base angles are equal, but they aren't vertical angles. So, does that mean the converse isn't always true?Wait, the question is just asking for the converse, not whether the converse is true or not. So, even if the converse isn't always true, it's still the correct converse of the original statement. I think that's right.Let me double-check my understanding. The original statement is "Vertical angles are equal," which is a true statement. Its converse would be "Equal angles are vertical angles," which, as I thought earlier, isn't necessarily true because equal angles can be formed in other ways besides being vertical angles.But for the purpose of this question, I'm only supposed to state the converse, not evaluate its truthfulness. So, I think I've got it. The converse is simply switching the hypothesis and conclusion of the original statement.Just to make sure, let me rephrase it. Original: If angles are vertical, then they are equal. Converse: If angles are equal, then they are vertical. Yeah, that seems to fit the definition of a converse.I guess another way to think about it is using symbols. Let's say V represents "angles are vertical" and E represents "angles are equal." The original statement is V → E. The converse would then be E → V. That aligns with what I concluded earlier.Okay, I feel confident now that the converse of "Vertical angles are equal" is "Equal angles are vertical angles." Even though the converse isn't always true, it's the correct logical converse of the original statement.

💡First, I will examine each sequence individually to identify the pattern.For the first sequence: 2, 4, 6, 8, 10, ______, ______- I notice that each term increases by 2.- To find the next two terms, I will add 2 to the last given term: 10 + 2 = 12, and then 12 + 2 = 14.For the second sequence: 3, 6, 12, 24, 48, ______, 192, …- I observe that each term is double the previous term.- To find the missing term, I will multiply the last given term by 2: 48 × 2 = 96.For the third sequence: 36, 11, 38, 13, 40, 15, 42, 17, ______, ______．- I notice that the odd-positioned terms increase by 2 each time: 36, 38, 40, 42, ...- The even-positioned terms also increase by 2 each time: 11, 13, 15, 17, ...- To find the next two terms, I will add 2 to the last odd and even terms respectively: 42 + 2 = 44, and 17 + 2 = 19.Finally, I will compile the answers based on the identified patterns.

💡Alright, so I have this trigonometric expression: cos x minus the square root of 3 times sin x. The problem is asking me to rewrite this expression in the form of 2 times sin(x plus some angle phi), where phi is between 0 and 2 pi. Then I need to figure out what phi is.Hmm, okay. I remember that there are formulas to combine sine and cosine terms into a single sine or cosine function. Maybe I can use one of those identities here. Let me think. I think it's something like a cos x plus b sin x can be written as R sin(x plus phi), where R is the amplitude and phi is the phase shift. Yeah, that sounds right.So, in this case, my expression is cos x minus sqrt(3) sin x. Comparing that to the general form a cos x plus b sin x, I can see that a is 1 and b is negative sqrt(3). So, a is 1, b is -sqrt(3). I think the formula for R is the square root of (a squared plus b squared). Let me calculate that. So, R would be sqrt(1^2 + (-sqrt(3))^2). That's sqrt(1 + 3), which is sqrt(4), so R is 2. Okay, that's good because the problem already has a 2 in front of the sine function, so that matches up.Now, to find phi, I think there's a relationship between a, b, and phi. I recall that cos phi is equal to a over R and sin phi is equal to b over R. Let me write that down:cos phi = a/R = 1/2sin phi = b/R = (-sqrt(3))/2So, cos phi is 1/2 and sin phi is negative sqrt(3)/2. I need to find an angle phi between 0 and 2 pi that satisfies both of these conditions.Let me think about the unit circle. Cos phi is positive, which means phi is in either the first or fourth quadrant. Sin phi is negative, so phi must be in the fourth quadrant. What angle in the fourth quadrant has a cosine of 1/2 and a sine of negative sqrt(3)/2? I know that cos(pi/3) is 1/2 and sin(pi/3) is sqrt(3)/2. But since we're in the fourth quadrant, the angle should be 2 pi minus pi/3, which is 5 pi/6. Wait, no, 2 pi minus pi/3 is 5 pi/3. Let me double-check that.Wait, pi/3 is 60 degrees, so 2 pi minus pi/3 is 360 minus 60, which is 300 degrees, which is 5 pi/3 radians. But wait, 5 pi/3 is in the fourth quadrant, and cos(5 pi/3) is 1/2, and sin(5 pi/3) is -sqrt(3)/2. That matches our conditions. So, phi is 5 pi/3.But hold on, the problem says to write it as 2 sin(x + phi). Let me make sure that's correct. If I use phi as 5 pi/3, then sin(x + 5 pi/3) would be the expression. But let me verify that.Alternatively, sometimes the formula is written as R sin(x + phi), where R is the amplitude and phi is the phase shift. So, let me recall the identity:a cos x + b sin x = R sin(x + phi)where R = sqrt(a^2 + b^2), and phi is such that cos phi = a/R and sin phi = b/R.Wait, but in this case, our expression is cos x - sqrt(3) sin x, which is a cos x + b sin x with a = 1 and b = -sqrt(3). So, yes, that should be equal to R sin(x + phi), where R is 2, as we found earlier.So, using the identity, we have:cos x - sqrt(3) sin x = 2 sin(x + phi)We found that cos phi = 1/2 and sin phi = -sqrt(3)/2, which gives phi = 5 pi/3.But let me double-check this by expanding 2 sin(x + 5 pi/3) and seeing if it equals cos x - sqrt(3) sin x.Using the sine addition formula:sin(x + 5 pi/3) = sin x cos(5 pi/3) + cos x sin(5 pi/3)We know that cos(5 pi/3) is 1/2 and sin(5 pi/3) is -sqrt(3)/2. So,sin(x + 5 pi/3) = sin x*(1/2) + cos x*(-sqrt(3)/2)Therefore, 2 sin(x + 5 pi/3) = 2*(sin x*(1/2) + cos x*(-sqrt(3)/2)) = sin x - sqrt(3) cos xWait, that's sin x - sqrt(3) cos x, but our original expression is cos x - sqrt(3) sin x. Hmm, that's different. So, that suggests that I might have made a mistake in determining phi.Let me think again. Maybe I confused the formula. Perhaps it's supposed to be R sin(x + phi) or R cos(x + phi). Let me check the identity again.I think the identity can be written in two ways: either as R sin(x + phi) or R cos(x + phi). Let me see which one applies here.If I use the identity for a cos x + b sin x, it can be written as R cos(x - phi), where R is sqrt(a^2 + b^2), and phi is such that cos phi = a/R and sin phi = b/R.Alternatively, it can also be written as R sin(x + theta), where theta is another angle. Maybe I need to adjust for that.Wait, let me see. Let me try writing it as R sin(x + phi). Then, expanding that:R sin(x + phi) = R sin x cos phi + R cos x sin phiComparing this to a cos x + b sin x, we have:a = R sin phib = R cos phiWait, that's different from what I thought earlier. So, in this case, a is 1, which is equal to R sin phi, and b is -sqrt(3), which is equal to R cos phi.So, R is 2, so:1 = 2 sin phi-sqrt(3) = 2 cos phiTherefore,sin phi = 1/2cos phi = -sqrt(3)/2So, now, sin phi is 1/2 and cos phi is -sqrt(3)/2. So, what angle phi satisfies this?Well, sin phi is positive and cos phi is negative, so phi is in the second quadrant.The reference angle for sin phi = 1/2 is pi/6. So, in the second quadrant, the angle is pi - pi/6 = 5 pi/6.So, phi is 5 pi/6.Wait, that's different from what I got earlier. So, earlier I thought phi was 5 pi/3, but that was when I was using the wrong identity. It seems that depending on how you set up the identity, phi can be different.Let me verify this by expanding 2 sin(x + 5 pi/6):sin(x + 5 pi/6) = sin x cos(5 pi/6) + cos x sin(5 pi/6)cos(5 pi/6) is -sqrt(3)/2 and sin(5 pi/6) is 1/2.So,sin(x + 5 pi/6) = sin x*(-sqrt(3)/2) + cos x*(1/2)Therefore, 2 sin(x + 5 pi/6) = 2*(-sqrt(3)/2 sin x + 1/2 cos x) = -sqrt(3) sin x + cos xWhich is exactly our original expression: cos x - sqrt(3) sin x.So, that works out. So, phi is 5 pi/6.Wait, so earlier I was confused because I was using the wrong identity, thinking that a cos x + b sin x equals R sin(x + phi), but actually, in that case, the coefficients correspond differently.So, to clarify, if I have a cos x + b sin x, and I want to write it as R sin(x + phi), then:R sin(x + phi) = R sin x cos phi + R cos x sin phiTherefore, equating coefficients:a = R sin phib = R cos phiSo, in our case, a is 1 and b is -sqrt(3). So,1 = R sin phi-sqrt(3) = R cos phiSince R is 2, we have:sin phi = 1/2cos phi = -sqrt(3)/2Which, as we saw, gives phi = 5 pi/6.Alternatively, if I had written it as R cos(x - phi), then:R cos(x - phi) = R cos x cos phi + R sin x sin phiSo, equating coefficients:a = R cos phib = R sin phiIn that case, with a = 1 and b = -sqrt(3):cos phi = 1/2sin phi = -sqrt(3)/2Which would give phi = 5 pi/3.So, depending on whether I write it as R sin(x + phi) or R cos(x - phi), the angle phi is different.But in the problem statement, it's specified to write it as 2 sin(x + phi). So, we need to use the identity that expresses it as a sine function with a phase shift.Therefore, in this case, phi is 5 pi/6.So, to summarize:We started with cos x - sqrt(3) sin x.We recognized that this can be written in the form R sin(x + phi), where R is 2.By expanding R sin(x + phi) and equating coefficients, we found that sin phi = 1/2 and cos phi = -sqrt(3)/2, leading us to phi = 5 pi/6.Therefore, the value of phi is 5 pi/6.

💡To determine the value of ( x ) such that the complex number ( (x - 2008) + (x + 2007)i ) is purely imaginary, we need to ensure that the real part of the complex number is zero.1. **Identify the Real Part:** The real part of the complex number is ( x - 2008 ).2. **Set the Real Part to Zero:** For the complex number to be purely imaginary, the real part must be zero. [ x - 2008 = 0 ]3. **Solve for ( x ):** [ x = 2008 ]4. **Verify the Imaginary Part:** The imaginary part is ( x + 2007 ). Substituting ( x = 2008 ): [ 2008 + 2007 = 4015 neq 0 ] This confirms that the imaginary part is non-zero, satisfying the condition for a purely imaginary number.Therefore, the value of ( x ) is ( 2008 ).