💡Okay, so I have this problem about table runners. Let me try to understand it step by step. There are five table runners, and their combined area is 360 square inches. They're overlapped to cover 90% of a table that's 250 square inches. So, first, I should figure out how much area they're covering on the table. 90% of 250 square inches is... let me calculate that. 0.9 times 250 is 225 square inches. So, the total area covered by the runners on the table is 225 square inches. Now, the problem also says that the area covered by exactly two layers of runner is 35 square inches, and the area covered by exactly four layers is 15 square inches. I need to find the area covered by exactly three layers. Hmm, okay. So, I think I need to use some principles of overlapping areas here. Maybe the principle of inclusion-exclusion? I remember that when calculating overlapping areas, you have to account for how many times each area is counted. Let me denote:- A1 as the area covered by exactly one layer,- A2 as the area covered by exactly two layers (which is given as 35),- A3 as the area covered by exactly three layers (what we need to find),- A4 as the area covered by exactly four layers (given as 15),- A5 as the area covered by all five layers, but since we have five runners, this might not be necessary unless specified.Wait, the problem doesn't mention the area covered by all five layers, so maybe it's zero or negligible. I'll assume it's zero for simplicity unless proven otherwise.So, the total area covered on the table is A1 + A2 + A3 + A4. We know that this total is 225 square inches. Plugging in the known values:A1 + 35 + A3 + 15 = 225So, A1 + A3 + 50 = 225Therefore, A1 + A3 = 175Okay, so the sum of areas covered by exactly one and exactly three layers is 175 square inches.Now, the combined area of all runners is 360 square inches. But when they overlap, the total area they cover on the table is less because of the overlaps. So, the total area of the runners is equal to the sum of the areas covered by each layer. That is:Total area of runners = A1 + 2*A2 + 3*A3 + 4*A4 + 5*A5But since we assumed A5 is zero, it simplifies to:360 = A1 + 2*35 + 3*A3 + 4*15Calculating the known values:2*35 = 704*15 = 60So, 360 = A1 + 70 + 3*A3 + 60Adding 70 and 60 gives 130, so:360 = A1 + 3*A3 + 130Subtracting 130 from both sides:230 = A1 + 3*A3But earlier, we found that A1 + A3 = 175So now we have two equations:1) A1 + A3 = 1752) A1 + 3*A3 = 230If I subtract equation 1 from equation 2, I can find A3:(A1 + 3*A3) - (A1 + A3) = 230 - 175Simplifying:2*A3 = 55So, A3 = 55 / 2 = 27.5Wait, that doesn't seem right. 27.5 square inches? But the problem is asking for an integer value, I think. Did I make a mistake somewhere?Let me check my calculations again.Total area covered on the table: 225 = A1 + 35 + A3 + 15 => A1 + A3 = 175Total area of runners: 360 = A1 + 2*35 + 3*A3 + 4*15Calculating 2*35 = 70 and 4*15 = 60, so 70 + 60 = 130Thus, 360 = A1 + 3*A3 + 130 => A1 + 3*A3 = 230Subtracting A1 + A3 = 175 from A1 + 3*A3 = 230:( A1 + 3*A3 ) - ( A1 + A3 ) = 230 - 1752*A3 = 55A3 = 27.5Hmm, 27.5 is 55/2, which is correct mathematically, but the problem might expect an integer. Maybe I made an incorrect assumption earlier.Wait, the problem doesn't specify that the area covered by five layers is zero. Maybe I shouldn't have assumed that. Let me consider A5 as a variable.So, revising the equations:Total area covered on the table: A1 + A2 + A3 + A4 + A5 = 225Given A2 = 35, A4 = 15, so:A1 + A3 + A5 = 225 - 35 - 15 = 175Total area of runners: A1 + 2*A2 + 3*A3 + 4*A4 + 5*A5 = 360Plugging in A2 = 35 and A4 = 15:A1 + 2*35 + 3*A3 + 4*15 + 5*A5 = 360Calculating:2*35 = 704*15 = 60So, 70 + 60 = 130Thus, A1 + 3*A3 + 5*A5 = 360 - 130 = 230Now, we have:1) A1 + A3 + A5 = 1752) A1 + 3*A3 + 5*A5 = 230Let me subtract equation 1 from equation 2:( A1 + 3*A3 + 5*A5 ) - ( A1 + A3 + A5 ) = 230 - 175Simplifying:2*A3 + 4*A5 = 55Divide both sides by 2:A3 + 2*A5 = 27.5Hmm, still getting a fractional value. Maybe A5 is zero? Let's test that.If A5 = 0, then A3 = 27.5But since the problem doesn't mention A5, perhaps it's zero. Alternatively, maybe the problem expects a fractional answer, which is unusual, but possible.Alternatively, perhaps I made a mistake in interpreting the total area covered. Let me double-check.Total area covered on the table is 90% of 250, which is 225. That's correct.Total area of runners is 360. That's given.When calculating the total area of runners, it's the sum of each layer's contribution. So, A1 is counted once, A2 is counted twice, A3 is counted thrice, etc.So, 360 = A1 + 2*A2 + 3*A3 + 4*A4 + 5*A5Yes, that's correct.Given A2 = 35 and A4 = 15, so 2*35 = 70 and 4*15 = 60, totaling 130.Thus, 360 - 130 = 230 = A1 + 3*A3 + 5*A5And from the total coverage, A1 + A3 + A5 = 175So, subtracting:230 - 175 = 55 = 2*A3 + 4*A5Which simplifies to A3 + 2*A5 = 27.5If A5 is zero, A3 = 27.5If A5 is non-zero, say A5 = x, then A3 = 27.5 - 2xBut since areas can't be negative, x must be less than or equal to 13.75But without more information, we can't determine x. So, perhaps the problem assumes A5 = 0, leading to A3 = 27.5But 27.5 is 55/2, which is 27.5 square inches. Maybe the answer is 27.5, but the problem might expect an integer. Alternatively, perhaps I made a mistake in the initial setup.Wait, let me check the total area covered again. The table is 250, covered 90%, so 225. The runners have a total area of 360. So, the overlapping areas account for 360 - 225 = 135 square inches of overlap.But how is this overlap distributed? The overlap is the sum of the areas covered by multiple layers beyond the first. So, for each area covered by two layers, it's an overlap of 1 (since it's counted twice in the total runner area). Similarly, areas covered by three layers have an overlap of 2, and so on.So, total overlap = A2 + 2*A3 + 3*A4 + 4*A5Given that total overlap is 135, and A2 = 35, A4 = 15, we have:35 + 2*A3 + 3*15 + 4*A5 = 135Calculating:35 + 45 = 80So, 80 + 2*A3 + 4*A5 = 135Thus, 2*A3 + 4*A5 = 55Which is the same as before: A3 + 2*A5 = 27.5So, same result. Therefore, unless A5 is non-zero, A3 is 27.5But the problem doesn't mention A5, so perhaps it's zero, and the answer is 27.5Alternatively, maybe I misread the problem. Let me check again.The problem says: "the area that is covered by exactly two layers of runner is 35 square inches and the area covered by exactly four layers of runner is 15 square inches."It doesn't mention areas covered by one or three layers, so we have to find the area covered by three layers.Given that, and the total coverage is 225, and the total runner area is 360, I think the calculations are correct, leading to A3 = 27.5But since the problem is presented in whole numbers, maybe I made a mistake in interpreting the total area.Wait, another approach: Let me use the principle of inclusion-exclusion for five sets.The formula for the union of five sets is:|A ∪ B ∪ C ∪ D ∪ E| = Σ|A| - Σ|A∩B| + Σ|A∩B∩C| - Σ|A∩B∩C∩D| + |A∩B∩C∩D∩E|But in this case, we're dealing with areas, not counts, so it's similar but with areas.Given that, the total area covered is:Total = A1 + A2 + A3 + A4 + A5Where A1 is area covered by exactly one runner, A2 by exactly two, etc.But we also know that the total area of all runners is:Σ|A| = 5* (area of each runner) = 360But since the runners are identical in area? Wait, no, the problem doesn't specify that they are identical. It just says five table runners with combined area 360. So, each runner could have different areas, but their total is 360.But without knowing individual areas, it's hard to apply inclusion-exclusion directly.Alternatively, perhaps the problem assumes that each runner has the same area, so each runner is 360/5 = 72 square inches.But the problem doesn't specify that, so I shouldn't assume that.Alternatively, maybe it's better to think in terms of the areas covered by different numbers of layers.Let me denote:Let x be the area covered by exactly one layer,y be the area covered by exactly two layers (given as 35),z be the area covered by exactly three layers (to find),w be the area covered by exactly four layers (given as 15),v be the area covered by all five layers.Then, the total area covered on the table is x + y + z + w + v = 225The total area of the runners is x + 2y + 3z + 4w + 5v = 360We have:1) x + y + z + w + v = 2252) x + 2y + 3z + 4w + 5v = 360Given y = 35, w = 15So, plugging in:1) x + 35 + z + 15 + v = 225 => x + z + v = 1752) x + 2*35 + 3z + 4*15 + 5v = 360 => x + 70 + 3z + 60 + 5v = 360 => x + 3z + 5v = 230Now, subtract equation 1 from equation 2:(x + 3z + 5v) - (x + z + v) = 230 - 175Simplifying:2z + 4v = 55Divide both sides by 2:z + 2v = 27.5So, z = 27.5 - 2vSince areas can't be negative, v must be ≤ 13.75But without more information, we can't determine v. So, unless v is zero, z is 27.5But the problem doesn't mention v, so perhaps v = 0, leading to z = 27.5Therefore, the area covered by exactly three layers is 27.5 square inches.But since the problem is presented with whole numbers, maybe I made a mistake in the setup.Wait, another thought: Maybe the total area covered is 225, and the total runner area is 360, so the overlapping area is 360 - 225 = 135This overlapping area is the sum of the overlaps from each layer beyond the first.So, for areas covered by two layers, the overlap is 35 (since each contributes an extra layer), for three layers, it's z (each contributes two extra layers), for four layers, it's 15 (each contributes three extra layers), and for five layers, it's v (each contributes four extra layers).So, total overlap = 35 + 2z + 3*15 + 4v = 35 + 2z + 45 + 4v = 80 + 2z + 4v = 135Thus, 2z + 4v = 55 => z + 2v = 27.5Same result as before.So, unless v is non-zero, z is 27.5But since the problem doesn't mention v, perhaps it's zero, making z = 27.5Alternatively, maybe the problem expects an integer, so perhaps I made a mistake in the initial setup.Wait, let me check the total area covered again. 90% of 250 is 225, correct.Total runner area is 360, correct.So, 360 = x + 2*35 + 3z + 4*15 + 5vWhich is x + 70 + 3z + 60 + 5v = x + 3z + 5v + 130 = 360Thus, x + 3z + 5v = 230And from the total coverage:x + z + v = 175Subtracting:2z + 4v = 55 => z + 2v = 27.5Same result.So, unless v is non-zero, z is 27.5But since the problem is presented with whole numbers, maybe I made a mistake in interpreting the problem.Wait, perhaps the area covered by exactly four layers is 15, but the area covered by five layers is also 15? No, the problem says exactly four layers is 15.Alternatively, maybe the area covered by five layers is zero, so z = 27.5But 27.5 is 55/2, which is correct mathematically.Alternatively, perhaps the problem expects the answer in fractional form, so 27.5 is acceptable.But in the initial problem, the assistant's answer was 65, which seems incorrect based on my calculations.Wait, let me check the assistant's answer.The assistant's answer was 65, but according to my calculations, it's 27.5Wait, that's a big discrepancy. Let me see where the assistant went wrong.Looking back:The assistant set up the equations as:X + 3Y + Z = 230X + 3Y = 95Then Z = 135Covered area = 225Then Y = 65Wait, that seems incorrect because:If X + 3Y = 95, and X + 3Y + Z = 230, then Z = 135But the covered area is 225, so 225 = X + 2*35 + 3Y + 4*15 = X + 70 + 3Y + 60 = X + 3Y + 130 = 95 + 130 = 225That's correct.But then the assistant subtracted 35 + 15 from 225 to get Y = 65But that's incorrect because the covered area is X + 2*35 + 3Y + 4*15 = 225Which is X + 70 + 3Y + 60 = 225 => X + 3Y = 95But the assistant then subtracted 35 + 15 from 225 to get Y = 65, which is incorrect because that approach doesn't account for the fact that X is also part of the covered area.So, the assistant's mistake was assuming that the covered area minus the areas covered by two and four layers gives the area covered by three layers, but that ignores the area covered by one layer.Therefore, the correct approach is to set up the equations as I did, leading to Y = 27.5But since the problem is presented with whole numbers, perhaps the correct answer is 65, but that contradicts the calculations.Alternatively, maybe the assistant made a mistake in the initial setup.Wait, let me try another approach.Let me denote:Let a = area covered by exactly one layer,b = area covered by exactly two layers = 35,c = area covered by exactly three layers,d = area covered by exactly four layers = 15,e = area covered by all five layers.Total area covered: a + b + c + d + e = 225Total runner area: a + 2b + 3c + 4d + 5e = 360Given b = 35, d = 15So,1) a + 35 + c + 15 + e = 225 => a + c + e = 1752) a + 2*35 + 3c + 4*15 + 5e = 360 => a + 70 + 3c + 60 + 5e = 360 => a + 3c + 5e = 230Subtracting equation 1 from equation 2:(a + 3c + 5e) - (a + c + e) = 230 - 175Simplifying:2c + 4e = 55 => c + 2e = 27.5So, c = 27.5 - 2eSince e must be a non-negative number, and c must be non-negative, e ≤ 13.75But without more information, we can't determine e, so c could be 27.5 if e = 0Therefore, the area covered by exactly three layers is 27.5 square inches.But the assistant's answer was 65, which seems incorrect based on this.Wait, perhaps the assistant made a mistake in the initial setup by not considering the area covered by one layer.In the assistant's solution, they set up:X + 3Y + Z = 230X + 3Y = 95Then Z = 135Covered area = 225Then Y = 65But this approach ignores the fact that the covered area is X + 2*35 + 3Y + 4*15 = 225Which is X + 70 + 3Y + 60 = 225 => X + 3Y = 95But then the assistant subtracted 35 + 15 from 225 to get Y = 65, which is incorrect because X is also part of the covered area.Therefore, the correct answer is 27.5 square inches, not 65.But since the problem is presented with whole numbers, perhaps the correct answer is 65, and I made a mistake in my calculations.Wait, let me try another approach.Let me denote:Let x be the area covered by exactly one layer,y = 35 (exactly two layers),z = ? (exactly three layers),w = 15 (exactly four layers),v = 0 (exactly five layers).Total area covered: x + y + z + w + v = x + 35 + z + 15 + 0 = x + z + 50 = 225 => x + z = 175Total runner area: x + 2y + 3z + 4w + 5v = x + 70 + 3z + 60 + 0 = x + 3z + 130 = 360 => x + 3z = 230Subtracting the first equation from the second:(x + 3z) - (x + z) = 230 - 175 => 2z = 55 => z = 27.5So, z = 27.5Therefore, the area covered by exactly three layers is 27.5 square inches.But the assistant's answer was 65, which is incorrect.So, the correct answer should be 27.5, but since the problem is presented with whole numbers, perhaps there's a mistake in the problem statement or the assistant's solution.Alternatively, maybe the assistant made a mistake in the initial setup.In conclusion, based on the calculations, the area covered by exactly three layers is 27.5 square inches.

💡First, I need to determine the total number of prizes inside the secret eggs. There are three categories of eggs based on the number of prizes they contain.For the eggs with 1 prize each:98 eggs × 1 prize per egg = 98 prizesFor the eggs with 2 prizes each:38 eggs × 2 prizes per egg = 76 prizesFor the eggs with 3 prizes each:9 eggs × 3 prizes per egg = 27 prizesAdding these together gives the total number of prizes:98 + 76 + 27 = 201 prizesNext, I'll add the total number of candies to the total number of prizes to find the total number of items in the jar:3409 candies + 201 prizes = 3610 itemsTherefore, the total number of items in the jar is 3610.

💡First, I need to calculate the value of ( x ) using the operation ( # ). The operation is defined as ( a # b = a + frac{a}{b} ).For ( x = 8 # 3 ):[x = 8 + frac{8}{3} = frac{24}{3} + frac{8}{3} = frac{32}{3}]Next, I calculate the value of ( y ) using the same operation.For ( y = 5 # 4 ):[y = 5 + frac{5}{4} = frac{20}{4} + frac{5}{4} = frac{25}{4}]Now, I need to find the product of ( x ) and ( y ):[x cdot y = frac{32}{3} times frac{25}{4} = frac{32 times 25}{3 times 4} = frac{800}{12} = frac{200}{3}]Therefore, the final answer is ( frac{200}{3} ).

💡First, I recognize that the ladder, the wall, and the ground form a right triangle. The ladder is the hypotenuse, measuring 13 meters, and the height it reaches up the wall is 12 meters, which is one of the legs.To find the distance from the base of the ladder to the wall, I can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.Applying the theorem, I set up the equation:13² = 12² + x²Calculating the squares:169 = 144 + x²Subtracting 144 from both sides:25 = x²Taking the square root of both sides:x = 5Therefore, the base of the ladder is 5 meters away from the wall.

💡First, I need to determine the total number of calories in the pan of brownies. Since there are 6 brownies and each contains 375 calories, I multiply 6 by 375 to get the total calories in the brownies.Next, I know that the cake has 526 more calories than the pan of brownies. So, I add 526 to the total calories of the brownies to find the total calories in the cake.Each slice of cake contains 347 calories. To find the number of slices in the cake, I divide the total calories of the cake by the calories per slice.Finally, performing the division gives me the number of slices in the cake.

💡Alright, so I've got this problem about a hyperbola, and I need to figure out which of the statements A, B, C, or D is true. Let me start by recalling what I know about hyperbolas.The equation given is (frac{x^2}{16} - frac{y^2}{9} = 1). I remember that the standard form of a hyperbola is (frac{(x - h)^2}{a^2} - frac{(y - k)^2}{b^2} = 1), where ((h, k)) is the center, (a) is the distance from the center to each vertex along the x-axis, and (b) is related to the conjugate axis. In this case, the center is at (0,0), (a^2 = 16), so (a = 4), and (b^2 = 9), so (b = 3).Next, I need to find the foci of the hyperbola. I remember that for hyperbolas, the distance from the center to each focus is given by (c), where (c^2 = a^2 + b^2). So, plugging in the values:(c^2 = 16 + 9 = 25), so (c = 5).Therefore, the foci (F_1) and (F_2) are located at ((-5, 0)) and ((5, 0)) respectively.Now, the problem states that the area of triangle (PF_1F_2) is 20. I need to use this information to determine which of the statements A, B, C, or D is true.Let me recall that the area of a triangle can be calculated using the formula:(text{Area} = frac{1}{2} times text{base} times text{height}).In this case, the base of the triangle is the distance between the two foci, which is (2c = 10). So, the base is 10 units. The height would be the distance from point (P) to the x-axis, since the foci are on the x-axis. Let's denote this height as (h). Therefore, the area is:(frac{1}{2} times 10 times h = 20).Solving for (h):(5h = 20) (h = 4).So, the distance from point (P) to the x-axis is 4. That means the y-coordinate of point (P) is either 4 or -4. Therefore, statement C is true because it says the distance from point (P) to the x-axis is 4.Now, let's check the other statements.Statement A: (|PF_1| - |PF_2| = 8).I remember that for hyperbolas, the definition is that the absolute difference of the distances from any point on the hyperbola to the two foci is constant and equal to (2a). Here, (a = 4), so (2a = 8). Therefore, (||PF_1| - |PF_2|| = 8). However, statement A says (|PF_1| - |PF_2| = 8), not the absolute value. Depending on which branch of the hyperbola point (P) is on, this difference could be positive or negative. So, without knowing the specific location of (P), we can't say for sure if (|PF_1| - |PF_2| = 8) or (|PF_2| - |PF_1| = 8). Therefore, statement A might not necessarily be true.Statement B: (|PF_1| + |PF_2| = frac{50}{3}).Hmm, this is about the sum of the distances from (P) to each focus. I know that for ellipses, the sum of distances is constant, but for hyperbolas, it's the difference that's constant. So, the sum isn't necessarily a fixed value. However, maybe we can find it using the information we have.We know the area of the triangle is 20, and we found that the y-coordinate of (P) is 4. Let's denote the coordinates of (P) as ((x, 4)). Since (P) lies on the hyperbola, it must satisfy the equation:(frac{x^2}{16} - frac{4^2}{9} = 1) (frac{x^2}{16} - frac{16}{9} = 1) (frac{x^2}{16} = 1 + frac{16}{9}) (frac{x^2}{16} = frac{25}{9}) (x^2 = frac{25}{9} times 16) (x^2 = frac{400}{9}) (x = pm frac{20}{3}).So, the x-coordinate of (P) is either (frac{20}{3}) or (-frac{20}{3}). Let's take (x = frac{20}{3}) first. Then, the coordinates of (P) are ((frac{20}{3}, 4)).Now, let's calculate (|PF_1|) and (|PF_2|).First, (F_1 = (-5, 0)) and (F_2 = (5, 0)).Calculating (|PF_1|):Distance between ((frac{20}{3}, 4)) and ((-5, 0)):(|PF_1| = sqrt{(frac{20}{3} + 5)^2 + (4 - 0)^2}) Convert 5 to thirds: (5 = frac{15}{3}), so:(|PF_1| = sqrt{(frac{20}{3} + frac{15}{3})^2 + 16}) (|PF_1| = sqrt{(frac{35}{3})^2 + 16}) (|PF_1| = sqrt{frac{1225}{9} + frac{144}{9}}) (|PF_1| = sqrt{frac{1369}{9}}) (|PF_1| = frac{37}{3}).Similarly, calculating (|PF_2|):Distance between ((frac{20}{3}, 4)) and ((5, 0)):(|PF_2| = sqrt{(frac{20}{3} - 5)^2 + (4 - 0)^2}) Convert 5 to thirds: (5 = frac{15}{3}), so:(|PF_2| = sqrt{(frac{20}{3} - frac{15}{3})^2 + 16}) (|PF_2| = sqrt{(frac{5}{3})^2 + 16}) (|PF_2| = sqrt{frac{25}{9} + frac{144}{9}}) (|PF_2| = sqrt{frac{169}{9}}) (|PF_2| = frac{13}{3}).Now, let's compute (|PF_1| + |PF_2|):(frac{37}{3} + frac{13}{3} = frac{50}{3}).So, statement B is true.If we take (x = -frac{20}{3}), the calculations would be similar, and we would find (|PF_1| + |PF_2| = frac{50}{3}) as well. So, regardless of which branch (P) is on, the sum is the same.Statement D: (∠F_1PF_2 = frac{pi}{3}).To check this, I can use the Law of Cosines in triangle (PF_1F_2). Let's denote the sides:- (|PF_1| = frac{37}{3}),- (|PF_2| = frac{13}{3}),- (|F_1F_2| = 10).The Law of Cosines states:(|F_1F_2|^2 = |PF_1|^2 + |PF_2|^2 - 2|PF_1||PF_2|cos(angle F_1PF_2)).Plugging in the values:(10^2 = left(frac{37}{3}right)^2 + left(frac{13}{3}right)^2 - 2 times frac{37}{3} times frac{13}{3} times cos(angle F_1PF_2)).Calculating each term:(100 = frac{1369}{9} + frac{169}{9} - 2 times frac{481}{9} cos(angle F_1PF_2)).Combine the fractions:(100 = frac{1369 + 169}{9} - frac{962}{9} cos(angle F_1PF_2)).(100 = frac{1538}{9} - frac{962}{9} cos(angle F_1PF_2)).Multiply both sides by 9 to eliminate denominators:(900 = 1538 - 962 cos(angle F_1PF_2)).Rearrange:(962 cos(angle F_1PF_2) = 1538 - 900).(962 cos(angle F_1PF_2) = 638).Divide both sides by 962:(cos(angle F_1PF_2) = frac{638}{962}).Simplify the fraction:Divide numerator and denominator by 2:(cos(angle F_1PF_2) = frac{319}{481}).Now, let's compute (frac{319}{481}) approximately:(319 ÷ 481 ≈ 0.663).Now, (cos(frac{pi}{3}) = 0.5), and (cos(frac{pi}{4}) ≈ 0.707). Since 0.663 is between 0.5 and 0.707, the angle is between (frac{pi}{3}) and (frac{pi}{4}). Therefore, (∠F_1PF_2) is not equal to (frac{pi}{3}). So, statement D is false.To recap:- Statement A: Not necessarily true because it depends on the branch of the hyperbola.- Statement B: True, as we calculated (|PF_1| + |PF_2| = frac{50}{3}).- Statement C: True, as the y-coordinate is 4.- Statement D: False, as the angle is not (frac{pi}{3}).Therefore, both statements B and C are true.**Final Answer**boxed{B text{ and } C}