answer：Okay, so Josh has this ribbon, right? He's got 18 yards of it. And he wants to use it equally for 6 gifts. Each gift is going to use 2 yards of ribbon. Hmm, I need to figure out how much ribbon will be left after he uses it for all the gifts.Alright, first, let's break this down. He has 18 yards total. He's going to use it for 6 gifts. Each gift uses 2 yards. So, maybe I should calculate how much ribbon he uses in total for all the gifts. That makes sense.So, if each gift uses 2 yards, and there are 6 gifts, then the total ribbon used would be 2 yards multiplied by 6 gifts. Let me write that down: 2 yards/gift * 6 gifts. That should give me the total ribbon used.Calculating that, 2 times 6 is 12. So, he uses 12 yards of ribbon in total for the gifts. Now, he started with 18 yards. To find out how much is left, I need to subtract the ribbon used from the total ribbon he had.So, that's 18 yards minus 12 yards. Let me do that subtraction: 18 - 12 equals 6. So, there should be 6 yards of ribbon left after he uses it for all the gifts.Wait, let me double-check that. If each gift uses 2 yards and there are 6 gifts, that's 12 yards used. Starting with 18 yards, subtracting 12 yards gives 6 yards remaining. Yeah, that seems right.I don't think I made any mistakes here. The calculations are straightforward: total ribbon used is 2 times 6, which is 12, and then subtracting that from 18 gives 6 yards left. So, the answer should be 6 yards of ribbon remaining.

answer：Okay, so I've got this geometry problem here, and I need to figure out what additional condition is required for line n to be perpendicular to plane β. Let me try to visualize and break it down step by step.First, the problem states that we have two planes, α and β, which are perpendicular to each other. Their intersection is line m. So, if I imagine two walls meeting at a right angle, the line where they meet is like line m. Now, there's another line, n, which is contained within plane α. So, line n is somewhere on one of those walls.The question is asking what extra condition we need to make sure that line n is perpendicular to plane β. The options are:A: m is parallel to n B: n is parallel to α C: n is perpendicular to m D: n is perpendicular to αHmm, okay. Let me recall some geometry concepts. I remember that if a line is perpendicular to a plane, it must be perpendicular to every line in that plane. But since n is in plane α, and α is already perpendicular to β, there must be some relationship between n and the intersection line m.Wait, so since α and β are perpendicular, any line in α that's perpendicular to m should also be perpendicular to β. Is that right? Let me think. If I have two perpendicular planes, then a line in one plane that's perpendicular to their intersection line should indeed be perpendicular to the other plane. That makes sense because it would be orthogonal to both planes.So, if line n is in plane α and is perpendicular to line m, which is the intersection of α and β, then n should be perpendicular to plane β. That seems to fit. So, the additional condition should be that n is perpendicular to m, which is option C.Let me check the other options to make sure I'm not missing something. Option A says m is parallel to n. If m and n were parallel, then n would just be another line in α parallel to the intersection. But being parallel doesn't necessarily make n perpendicular to β. In fact, it might not even be perpendicular at all. So, A doesn't seem right.Option B says n is parallel to α. Wait, n is already in α, so being parallel to α doesn't add any new information. It's redundant because any line in α is already parallel to α in a way. So, B isn't the condition we need.Option D says n is perpendicular to α. But if n is perpendicular to α, that would mean n is a line sticking out of α, but the problem states that n is a subset of α. So, n can't be perpendicular to α because it's lying entirely within α. That doesn't make sense. So, D is out.Therefore, the only logical choice is C: n is perpendicular to m. That makes n perpendicular to plane β because it satisfies the condition required by the theorem.I think I've got it. The key was remembering how lines and planes interact when they're perpendicular. Since α and β are perpendicular, their intersection m is like the axis around which their perpendicularity is defined. So, if a line in α is perpendicular to m, it must be sticking straight out of β as well, making it perpendicular to β.Yeah, I feel confident now. Option C is the one we need.

answer：First, I need to determine the initial number of pencils Anthony had.I know that Anthony gave Kathryn 9.0 pencils and was left with 47 pencils.To find the initial number of pencils, I can add the number of pencils he gave away to the number of pencils he had left.So, the calculation would be:9.0 (pencils given to Kathryn) + 47 (pencils left with Anthony) = 56.0 pencilsTherefore, Anthony initially had 56 pencils.

answer：First, I need to determine the number of marbles in sack (B). Since there are a total of 72 marbles in both sacks and sack (A) contains 42 marbles, I can subtract the number of marbles in sack (A) from the total to find the number in sack (B).Next, to find out how many more marbles are in sack (A) compared to sack (B), I subtract the number of marbles in sack (B) from the number of marbles in sack (A).This calculation will give me the difference in the number of marbles between the two sacks.

answer：Alright, so I have this problem here: Given the function f(x) = (1/2)^x - 3, I need to find a necessary but not sufficient condition for f(x) < 5. The options are A: x > -4, B: x > -3, C: x < -2, and D: x < -3.First, I need to understand what the question is asking. A necessary condition means that for f(x) < 5 to be true, this condition must hold. However, it's not sufficient, meaning that just having this condition doesn't guarantee that f(x) < 5. So, I need to find an inequality that must be true whenever f(x) < 5, but there are cases where this inequality is true, but f(x) is not less than 5.Okay, let's start by solving the inequality f(x) < 5. The function is f(x) = (1/2)^x - 3, so:(1/2)^x - 3 < 5I can add 3 to both sides to isolate the exponential term:(1/2)^x < 8Now, I need to solve for x. I know that (1/2)^x is the same as 2^(-x). So, rewriting the inequality:2^(-x) < 8I also know that 8 is 2^3, so:2^(-x) < 2^3Since the bases are the same and greater than 1, the inequality of the exponents will hold:-x < 3Multiplying both sides by -1 (and remembering to flip the inequality sign):x > -3So, x must be greater than -3 for f(x) < 5 to hold. That means x > -3 is a necessary condition because if f(x) < 5, then x must be greater than -3. However, is it sufficient? Let's see.If x > -3, does that guarantee f(x) < 5? Let's test x = -2, which is greater than -3.f(-2) = (1/2)^(-2) - 3 = 2^2 - 3 = 4 - 3 = 1 < 5Okay, that works. How about x = 0?f(0) = (1/2)^0 - 3 = 1 - 3 = -2 < 5Still works. What about x approaching infinity?As x increases, (1/2)^x approaches 0, so f(x) approaches -3, which is still less than 5.Wait, so if x > -3, f(x) is always less than 5. That means x > -3 is both necessary and sufficient. But the question asks for a necessary but not sufficient condition. So, x > -3 is not the answer because it's sufficient as well.Looking back at the options, A is x > -4. Is x > -4 a necessary condition? Let's see. If f(x) < 5, does x have to be greater than -4?From earlier, we know that x must be greater than -3 for f(x) < 5. So, if x is greater than -3, it's automatically greater than -4. Therefore, x > -4 is a necessary condition because whenever f(x) < 5, x must be greater than -4. But is it sufficient?If x > -4, does that guarantee f(x) < 5? Let's test x = -3.5, which is greater than -4 but less than -3.f(-3.5) = (1/2)^(-3.5) - 3 = 2^3.5 - 3 ≈ 11.31 - 3 ≈ 8.31 > 5So, even though x > -4, f(x) can be greater than 5. Therefore, x > -4 is not sufficient, but it is necessary.So, the answer should be A: x > -4.Wait, but let me double-check. If x > -4 is necessary, does that mean that whenever f(x) < 5, x must be greater than -4? Yes, because if x were less than or equal to -4, then x would be less than -3, and we know that for f(x) < 5, x must be greater than -3. So, x > -4 is indeed a necessary condition.Is there any other option that could be necessary but not sufficient? Let's look at the other options.Option B is x > -3, which we saw is both necessary and sufficient, so it's not the answer.Option C is x < -2. Is this a necessary condition? Let's see. If f(x) < 5, does x have to be less than -2? No, because x can be greater than -3, which includes values greater than -2 as well. So, x < -2 is not necessary.Option D is x < -3. Similarly, if f(x) < 5, x must be greater than -3, so x < -3 is not necessary.Therefore, the only necessary but not sufficient condition is A: x > -4.

answer：First, I need to define the height functions for both plants as functions of time ( t ) (in years).For the first plant:- Current height: 44 cm- Growth rate: 3 cm every 2 years- Growth per year: (frac{3}{2}) cm/year- Height function: ( h_1(t) = 44 + frac{3}{2}t )For the second plant:- Current height: 80 cm- Growth rate: 5 cm every 6 years- Growth per year: (frac{5}{6}) cm/year- Height function: ( h_2(t) = 80 + frac{5}{6}t )Next, I set the two height functions equal to each other to find the time ( t ) when both plants are the same height:[44 + frac{3}{2}t = 80 + frac{5}{6}t]To solve for ( t ), I first subtract 44 from both sides:[frac{3}{2}t = 36 + frac{5}{6}t]Then, subtract (frac{5}{6}t) from both sides:[frac{3}{2}t - frac{5}{6}t = 36]Finding a common denominator for the fractions:[frac{9}{6}t - frac{5}{6}t = 36]Combining the fractions:[frac{4}{6}t = 36]Simplifying the fraction:[frac{2}{3}t = 36]Multiplying both sides by (frac{3}{2}):[t = 36 cdot frac{3}{2} = 54]Finally, I verify the solution by calculating the heights of both plants after 54 years:- First plant: ( 44 + frac{3}{2} cdot 54 = 125 ) cm- Second plant: ( 80 + frac{5}{6} cdot 54 = 125 ) cmBoth plants are indeed 125 cm tall after 54 years, confirming the solution.