answer：Okay, so I have this problem where I need to find the flux of a vector field through the upper side of a triangle ABC. The vertices of the triangle are given as A(1,0,0), B(0,1,0), and C(0,0,1). The vector field is **a** = (x - 2z)i + (x + 3y + z)j + (5x + y)k. Hmm, flux through a surface... I remember that flux is calculated using a surface integral of the vector field dotted with the normal vector to the surface. First, I need to figure out the equation of the plane that forms the triangle ABC. Since the triangle is in 3D space, I can find the equation of the plane by using the coordinates of the three points. Let me recall that the general equation of a plane is ax + by + cz + d = 0. To find the coefficients a, b, c, I can use the normal vector to the plane, which can be found by taking the cross product of two vectors lying on the plane.So, let me define two vectors on the plane. Vector AB would be from point A to point B, which is B - A = (0-1, 1-0, 0-0) = (-1, 1, 0). Similarly, vector AC is from point A to point C, which is C - A = (0-1, 0-0, 1-0) = (-1, 0, 1). Now, I can compute the cross product of AB and AC to get the normal vector.Calculating the cross product AB × AC:AB = (-1, 1, 0)AC = (-1, 0, 1)The cross product is given by the determinant:|i   j   k||-1  1   0||-1  0   1|So, i*(1*1 - 0*0) - j*(-1*1 - 0*(-1)) + k*(-1*0 - 1*(-1)).Calculating each component:i*(1 - 0) = i*1 = i-j*(-1 - 0) = -j*(-1) = jk*(0 + 1) = k*1 = kSo, the cross product AB × AC = i + j + k. Therefore, the normal vector is (1, 1, 1). Now, the equation of the plane can be written using point A(1,0,0). The general form is 1(x - 1) + 1(y - 0) + 1(z - 0) = 0, which simplifies to x - 1 + y + z = 0, so x + y + z = 1. That's the equation of the plane.Next, I need to parameterize the surface of the triangle. Since it's a triangle in the plane x + y + z = 1, I can use two parameters, say u and v, to express the coordinates x, y, z. Alternatively, I can express z in terms of x and y, since the plane equation allows that. From x + y + z = 1, we get z = 1 - x - y. So, if I consider x and y as parameters, then the surface can be parameterized as:r(x, y) = x i + y j + (1 - x - y) kBut I need to define the limits for x and y such that the points lie on the triangle ABC. The triangle has vertices at (1,0,0), (0,1,0), and (0,0,1). So, in the x-y plane, the projection of the triangle would be a triangle with vertices at (1,0), (0,1), and (0,0). Therefore, x and y must satisfy x ≥ 0, y ≥ 0, and x + y ≤ 1.So, the parameterization is valid for x from 0 to 1, and for each x, y from 0 to 1 - x.Now, to compute the flux, I need to set up the surface integral:Flux = ∫∫_S **a** · d**S**Where d**S** is the vector differential element of the surface, which is equal to **n** dS, where **n** is the unit normal vector. Alternatively, in terms of parameterization, d**S** can be expressed as (r_x × r_y) dx dy, where r_x and r_y are the partial derivatives of r with respect to x and y.So, let's compute the partial derivatives:r_x = ∂r/∂x = i + 0j + (-1)k = i - kr_y = ∂r/∂y = 0i + j + (-1)k = j - kNow, compute the cross product r_x × r_y:|i   j   k||1    0  -1||0    1  -1|Calculating the determinant:i*(0*(-1) - (-1)*1) - j*(1*(-1) - (-1)*0) + k*(1*1 - 0*0)Simplify each component:i*(0 + 1) = i*1 = i-j*(-1 - 0) = -j*(-1) = jk*(1 - 0) = k*1 = kSo, r_x × r_y = i + j + k, which is the same as the normal vector we found earlier. That makes sense because the cross product of the partial derivatives gives the normal vector to the surface.But wait, I need to make sure about the orientation. The problem mentions the "upper side" of the triangle. In 3D, the orientation matters because the normal vector can point in two directions. Since the cross product r_x × r_y gives a normal vector pointing in the direction of (1,1,1), which is upwards in this case because the z-component is positive. So, that should be the correct orientation for the upper side.Therefore, d**S** = (i + j + k) dx dy.Now, the flux integral becomes:Flux = ∫∫_D **a** · (i + j + k) dx dyWhere D is the projection of the surface onto the x-y plane, which is the triangle with vertices (1,0), (0,1), (0,0).So, let's compute the dot product **a** · (i + j + k). The vector field **a** is given as (x - 2z)i + (x + 3y + z)j + (5x + y)k. So, substituting z from the plane equation, z = 1 - x - y.Therefore, let's express each component of **a** in terms of x and y:First component (i): x - 2z = x - 2(1 - x - y) = x - 2 + 2x + 2y = 3x + 2y - 2Second component (j): x + 3y + z = x + 3y + (1 - x - y) = x + 3y + 1 - x - y = 2y + 1Third component (k): 5x + ySo, **a** = (3x + 2y - 2)i + (2y + 1)j + (5x + y)kNow, compute the dot product with (i + j + k):**a** · (i + j + k) = (3x + 2y - 2) + (2y + 1) + (5x + y)Simplify term by term:First term: 3x + 2y - 2Second term: 2y + 1Third term: 5x + yAdding them together:3x + 2y - 2 + 2y + 1 + 5x + yCombine like terms:x terms: 3x + 5x = 8xy terms: 2y + 2y + y = 5yconstants: -2 + 1 = -1So, the dot product simplifies to 8x + 5y - 1.Therefore, the flux integral becomes:Flux = ∫∫_D (8x + 5y - 1) dx dyWhere D is the triangular region x ≥ 0, y ≥ 0, x + y ≤ 1.Now, I need to set up the double integral over D. Since it's a triangle, I can set up the limits as x from 0 to 1, and for each x, y from 0 to 1 - x.So, the integral becomes:Flux = ∫_{x=0}^{1} ∫_{y=0}^{1 - x} (8x + 5y - 1) dy dxLet me compute the inner integral first with respect to y:∫_{y=0}^{1 - x} (8x + 5y - 1) dyLet's integrate term by term:Integral of 8x dy = 8x * y evaluated from 0 to 1 - x = 8x*(1 - x) - 8x*0 = 8x(1 - x)Integral of 5y dy = (5/2)y² evaluated from 0 to 1 - x = (5/2)(1 - x)^2 - 0 = (5/2)(1 - 2x + x²)Integral of (-1) dy = -1 * y evaluated from 0 to 1 - x = -(1 - x) - 0 = - (1 - x)So, putting it all together:Inner integral = 8x(1 - x) + (5/2)(1 - 2x + x²) - (1 - x)Let me expand each term:First term: 8x - 8x²Second term: (5/2)(1) - (5/2)(2x) + (5/2)(x²) = 5/2 - 5x + (5/2)x²Third term: -1 + xNow, combine all terms:8x - 8x² + 5/2 - 5x + (5/2)x² -1 + xCombine like terms:x terms: 8x -5x + x = 4xx² terms: -8x² + (5/2)x² = (-16/2 + 5/2)x² = (-11/2)x²constants: 5/2 -1 = 5/2 - 2/2 = 3/2So, the inner integral simplifies to:4x - (11/2)x² + 3/2Now, the flux integral becomes:Flux = ∫_{x=0}^{1} [4x - (11/2)x² + 3/2] dxLet's integrate term by term:Integral of 4x dx = 2x²Integral of -(11/2)x² dx = -(11/6)x³Integral of 3/2 dx = (3/2)xSo, putting it all together:Flux = [2x² - (11/6)x³ + (3/2)x] evaluated from 0 to 1Compute at x = 1:2(1)^2 - (11/6)(1)^3 + (3/2)(1) = 2 - 11/6 + 3/2Convert all terms to sixths:2 = 12/611/6 = 11/63/2 = 9/6So:12/6 - 11/6 + 9/6 = (12 - 11 + 9)/6 = 10/6 = 5/3Compute at x = 0:All terms are zero, so the result is 0.Therefore, the flux is 5/3 - 0 = 5/3.Wait, hold on. Let me double-check my calculations because I might have made a mistake somewhere.Starting from the inner integral:∫ (8x + 5y - 1) dy from 0 to 1 - xI had:8x(1 - x) + (5/2)(1 - x)^2 - (1 - x)Wait, let me recompute that step because I think I might have messed up the expansion.Wait, no, the expansion was correct. Let me re-express the inner integral:8x(1 - x) = 8x - 8x²(5/2)(1 - 2x + x²) = 5/2 - 5x + (5/2)x²-(1 - x) = -1 + xSo, adding all terms:8x - 8x² + 5/2 -5x + (5/2)x² -1 + xCombine x terms: 8x -5x + x = 4xx² terms: -8x² + (5/2)x² = (-16/2 + 5/2)x² = (-11/2)x²constants: 5/2 -1 = 3/2So, yes, that part is correct.Then, integrating 4x - (11/2)x² + 3/2 with respect to x:Integral of 4x is 2x²Integral of -(11/2)x² is -(11/6)x³Integral of 3/2 is (3/2)xSo, evaluating from 0 to 1:2(1)^2 - (11/6)(1)^3 + (3/2)(1) = 2 - 11/6 + 3/2Convert to sixths:2 = 12/611/6 = 11/63/2 = 9/6So, 12/6 - 11/6 + 9/6 = (12 -11 +9)/6 = 10/6 = 5/3Yes, that seems correct.Wait, but I just thought, is the normal vector pointing in the correct direction? Because sometimes when parameterizing, the cross product can give a normal vector pointing in the opposite direction. But in this case, since the cross product r_x × r_y gave us (1,1,1), which has positive z-component, and since the triangle is in the upper side, meaning it's oriented such that the normal vector points upwards, which is consistent with the cross product we computed. So, the orientation is correct.Therefore, the flux should be 5/3.Wait, but let me think again. The problem says "the upper side of the triangle ABC." So, if the normal vector is pointing upwards, which is consistent with our calculation, then 5/3 should be the correct answer.Alternatively, maybe I should compute the flux using the divergence theorem? But wait, the divergence theorem applies to closed surfaces, and here we're dealing with an open surface, a triangle. So, that might not be applicable here.Alternatively, another way to compute flux is to use the formula:Flux = ∫∫_S **a** · **n** dSWhere **n** is the unit normal vector. Since we already have the normal vector as (1,1,1), the unit normal vector would be (1,1,1) divided by its magnitude. The magnitude is sqrt(1^2 + 1^2 + 1^2) = sqrt(3). So, **n** = (1/sqrt(3), 1/sqrt(3), 1/sqrt(3)).But in our earlier approach, we used the cross product r_x × r_y, which gave us the non-unit normal vector (1,1,1). So, when we computed the flux as ∫∫ **a** · (r_x × r_y) dx dy, that already accounts for the scaling factor, so we don't need to worry about the unit normal vector in that case. So, I think my initial approach is correct.Therefore, I think the flux is indeed 5/3.But just to be thorough, let me recompute the integral step by step.First, the parameterization:r(x, y) = x i + y j + (1 - x - y) kPartial derivatives:r_x = i - kr_y = j - kCross product:r_x × r_y = |i   j   k|             1    0   -1             0    1   -1= i*(0*(-1) - (-1)*1) - j*(1*(-1) - (-1)*0) + k*(1*1 - 0*0)= i*(0 +1) - j*(-1 -0) + k*(1 -0)= i + j + kSo, that's correct.Then, **a** · (i + j + k):**a** = (x - 2z)i + (x + 3y + z)j + (5x + y)kSubstitute z = 1 - x - y:First component: x - 2(1 - x - y) = x - 2 + 2x + 2y = 3x + 2y - 2Second component: x + 3y + (1 - x - y) = x + 3y +1 -x - y = 2y +1Third component: 5x + yDot product: (3x + 2y -2) + (2y +1) + (5x + y) = 8x +5y -1Yes, that's correct.Then, setting up the integral over D:∫_{0}^{1} ∫_{0}^{1 -x} (8x +5y -1) dy dxCompute inner integral:∫ (8x +5y -1) dy = 8x y + (5/2)y² - y evaluated from 0 to 1 -xAt y =1 -x:8x(1 -x) + (5/2)(1 -x)^2 - (1 -x)At y=0: 0 +0 -0 =0So, the inner integral is:8x(1 -x) + (5/2)(1 -2x +x²) - (1 -x)Expand:8x -8x² + 5/2 -5x + (5/2)x² -1 +xCombine like terms:x terms: 8x -5x +x =4xx² terms: -8x² + (5/2)x² = (-16/2 +5/2)x² = (-11/2)x²constants: 5/2 -1 =3/2So, inner integral is 4x - (11/2)x² + 3/2Now, integrate with respect to x:∫ (4x - (11/2)x² + 3/2) dx from 0 to1Compute term by term:∫4x dx = 2x²∫-(11/2)x² dx = -(11/6)x³∫3/2 dx = (3/2)xSo, the integral becomes:[2x² - (11/6)x³ + (3/2)x] from 0 to1At x=1:2(1) - (11/6)(1) + (3/2)(1) =2 -11/6 +3/2Convert to sixths:2=12/6, 11/6=11/6, 3/2=9/6So, 12/6 -11/6 +9/6= (12 -11 +9)/6=10/6=5/3At x=0, all terms are zero.Thus, flux=5/3.Yes, that seems consistent. I think I was thorough in checking each step, so I feel confident that the flux is indeed 5/3.**Final Answer**The flux of the vector field through the upper side of the triangle ABC is boxed{dfrac{5}{3}}.

answer：Okay, so I have this equation here: (m + 2)x^{|m| - 1} + 8 = 0. And I need to find the value of m such that this equation is linear. Hmm, linear equations are of the form ax + b = 0, right? So, the exponent on x has to be 1. That means the exponent |m| - 1 should equal 1. Let me write that down.So, |m| - 1 = 1. Solving for |m|, I add 1 to both sides: |m| = 2. That means m can be either 2 or -2 because absolute value of both 2 and -2 is 2. Okay, so m is either 2 or -2. But wait, I need to check if both of these make the equation linear or if one of them causes a problem.Let me substitute m = 2 into the equation. Then, (2 + 2)x^{|2| - 1} + 8 = 0 simplifies to 4x^{2 - 1} + 8 = 0, which is 4x^1 + 8 = 0, so 4x + 8 = 0. That's definitely a linear equation. Good.Now, let me try m = -2. Substituting that in, we get (-2 + 2)x^{|-2| - 1} + 8 = 0. Simplifying, (-2 + 2) is 0, so the equation becomes 0 * x^{2 - 1} + 8 = 0, which is 0x + 8 = 0, or just 8 = 0. Wait, that's not possible. 8 doesn't equal 0, so this equation doesn't hold true. So, m = -2 would make the equation invalid because it results in a false statement.Therefore, m = -2 is not a valid solution because it doesn't give a linear equation but instead an impossible equation. So, the only valid solution is m = 2.Let me just double-check my steps to make sure I didn't make a mistake. First, I set the exponent equal to 1 because it's a linear equation, so |m| - 1 = 1, leading to |m| = 2. Then, I considered both m = 2 and m = -2. Substituting m = 2 works, giving a linear equation, but substituting m = -2 results in 8 = 0, which is not true. So, yes, m must be 2.I think that's solid. I don't see any mistakes in my reasoning. So, the answer should be m = 2.**Final Answer**The value of ( m ) is boxed{2}.

answer：Okay, so I have this problem where I need to take any 3-digit number, say abc, and then enter it again on a calculator to make a 6-digit number, which would be abcabc. Then, I have to divide this number by 7, then by 11, and finally by 13. The question is asking what happens and why. Hmm, interesting.Let me break this down step by step. First, let's represent the 3-digit number as abc, where a, b, c are digits. So, mathematically, the number can be written as 100a + 10b + c. When I enter this number again, it becomes a 6-digit number, which is abcabc. So, how do I represent this 6-digit number?Well, if I think about it, the first three digits are abc, which is 100a + 10b + c, and the next three digits are also abc. But since the next three digits are shifted three places to the left, it's actually (100a + 10b + c) multiplied by 1000. So, the entire 6-digit number would be:(100a + 10b + c) * 1000 + (100a + 10b + c)Which simplifies to:(100a + 10b + c) * (1000 + 1) = (100a + 10b + c) * 1001Oh, so the 6-digit number is just the original 3-digit number multiplied by 1001. That's a key insight. So, abcabc = abc * 1001.Now, the problem says to divide this number by 7, then by 11, and then by 13. Let me write that out:abcabc / 7 / 11 / 13But since division is associative, I can rewrite this as:abcabc / (7 * 11 * 13)Let me compute 7 * 11 * 13. 7 times 11 is 77, and 77 times 13. Let me calculate that:77 * 10 = 77077 * 3 = 231So, 770 + 231 = 1001Oh! So, 7 * 11 * 13 equals 1001. Therefore, the expression becomes:abcabc / 1001But earlier, I established that abcabc is equal to abc * 1001. So, substituting that in:(abc * 1001) / 1001 = abcSo, the result of dividing abcabc by 7, then by 11, and then by 13 is just the original 3-digit number, abc.Wait, so does that mean that no matter what 3-digit number I choose, when I perform these divisions, I always end up with the original number? That seems to be the case.Let me test this with an example to make sure I didn't make a mistake. Let's pick a random 3-digit number, say 123. Then, entering it again gives 123123.Dividing 123123 by 7: 123123 ÷ 7. Let me compute that.7 goes into 12 once, remainder 5. 7 into 53 (5 from remainder and 3) is 7, remainder 4. 7 into 41 is 5, remainder 6. 7 into 62 is 8, remainder 6. 7 into 63 is 9, remainder 0. So, 123123 ÷ 7 is 17589.Then, divide that by 11: 17589 ÷ 11. Let's see.11 into 17 is 1, remainder 6. 11 into 65 is 5, remainder 10. 11 into 108 is 9, remainder 9. 11 into 99 is 9, remainder 0. So, 17589 ÷ 11 is 1599.Wait, hold on, that doesn't seem right. 11 times 1599 is 17589, correct. But then, dividing 1599 by 13.13 into 15 is 1, remainder 2. 13 into 29 is 2, remainder 3. 13 into 39 is 3, remainder 0. So, 1599 ÷ 13 is 123. Which is the original number. So, that works.Wait, but when I did 123123 ÷ 7, I got 17589, then ÷11 is 1599, then ÷13 is 123. So, it worked.Let me try another number, maybe 456. So, 456456.Divide by 7: 456456 ÷ 7. Let's compute that.7 into 45 is 6, remainder 3. 7 into 36 is 5, remainder 1. 7 into 14 is 2, remainder 0. 7 into 5 is 0, remainder 5. 7 into 56 is 8, remainder 0. So, 456456 ÷ 7 is 65208.Then, divide by 11: 65208 ÷ 11.11 into 65 is 5, remainder 10. 11 into 102 is 9, remainder 3. 11 into 30 is 2, remainder 8. 11 into 88 is 8, remainder 0. So, 65208 ÷ 11 is 5928.Then, divide by 13: 5928 ÷ 13.13 into 59 is 4, remainder 7. 13 into 72 is 5, remainder 7. 13 into 78 is 6, remainder 0. So, 5928 ÷ 13 is 456. Which is the original number. So, that works too.Wait, so it seems consistent. So, regardless of the 3-digit number, when you create the 6-digit number by repeating it, and then divide by 7, 11, and 13 in succession, you get back the original number.So, why does this happen? It's because 1001 is the product of 7, 11, and 13. So, when you create the 6-digit number, you're effectively multiplying the original 3-digit number by 1001. Then, dividing by 7, 11, and 13 is equivalent to dividing by 1001, which cancels out the multiplication, leaving you with the original number.Therefore, the result is the original 3-digit number, and this happens because 7 × 11 × 13 equals 1001, which is the factor by which the original number was multiplied when creating the 6-digit number.I think that's the reasoning. Let me just recap to make sure I didn't miss anything. The key steps are:1. Representing the 6-digit number as the original 3-digit number multiplied by 1001.2. Recognizing that 1001 factors into 7 × 11 × 13.3. Therefore, dividing by these three numbers in succession is equivalent to dividing by 1001, which undoes the multiplication, leaving the original number.Yes, that makes sense. I don't see any flaws in this reasoning. The example with 123 and 456 both worked, so it seems consistent.**Final Answer**The result is the original 3-digit number, so the final answer is boxed{abc}.Wait, but in the problem statement, they didn't specify to write the original number, but rather to explain what happened. So, perhaps the box should contain the original number, but since it's variable, maybe just stating that it results in the original number. Alternatively, if the question is asking for the result, which is the original number, but since it's variable, perhaps the answer is that it results in the original number, which can be represented as boxed{abc}.Alternatively, if they expect a numerical answer, but since it's any 3-digit number, it's variable. So, the answer is that you get back the original 3-digit number, so boxed{abc} is appropriate.**Final Answer**boxed{abc}

answer：To determine which day the public holiday cannot fall on, I'll analyze the possible dates for the third Wednesday of a month.First, I'll consider the earliest possible starting day for the month. If the first day of the month is a Sunday, the Wednesdays would fall on the 2nd, 9th, 16th, and 23rd. In this case, the third Wednesday is the 16th.Next, if the first day is a Monday, the Wednesdays would be on the 1st, 8th, 15th, and 22nd. Here, the third Wednesday is the 15th.If the first day is a Tuesday, the Wednesdays would be on the 7th, 14th, 21st, and 28th. The third Wednesday is the 21st.If the first day is a Wednesday, the Wednesdays would be on the 6th, 13th, 20th, and 27th. The third Wednesday is the 20th.If the first day is a Thursday, the Wednesdays would be on the 5th, 12th, 19th, and 26th. The third Wednesday is the 19th.If the first day is a Friday, the Wednesdays would be on the 4th, 11th, 18th, and 25th. The third Wednesday is the 18th.Lastly, if the first day is a Saturday, the Wednesdays would be on the 3rd, 10th, 17th, and 24th. The third Wednesday is the 17th.From this analysis, the possible dates for the third Wednesday are the 15th, 16th, 17th, 18th, 19th, 20th, 21st, and 22nd. The 23rd is not among these possible dates. Therefore, the holiday cannot occur on the 23rd.

answer：First, Billy sleeps 6 hours on the first night.On the second night, he sleeps 2 more hours than the first night, which is 6 + 2 = 8 hours.On the third night, he sleeps half of the previous night's amount, so 8 / 2 = 4 hours.Finally, on the fourth night, he sleeps triple the amount of the third night, which is 4 * 3 = 12 hours.Adding up all the hours: 6 + 8 + 4 + 12 = 30 hours.

answer：Okay, so I have this problem here: If the ceiling of the square root of x is equal to 15, how many possible integer values of x are there? Hmm, let me think about how to approach this.First, I remember that the ceiling function, denoted by ⎡·⎤, rounds a number up to the nearest integer. So, if ⎡√x⎤ = 15, that means that √x is somewhere between 14 and 15, but not reaching 15. Wait, is that right? Let me verify.If the ceiling of a number is 15, then the number must be greater than 14 but less than or equal to 15. Because if it were exactly 15, the ceiling would still be 15, right? So, actually, √x must satisfy 14 < √x ≤ 15.But hold on, is that correct? Let me think again. The ceiling function rounds up to the smallest integer greater than or equal to the number. So, if √x is exactly 14, then the ceiling would be 14, not 15. Therefore, for the ceiling to be 15, √x must be greater than 14 but less than or equal to 15. So, 14 < √x ≤ 15.Okay, that makes sense. So, if I square all parts of this inequality, I can find the range of x. Let's do that.Starting with 14 < √x ≤ 15.Squaring the left part: 14² = 196.Squaring the middle: (√x)² = x.Squaring the right part: 15² = 225.So, putting it all together, 196 < x ≤ 225.But wait, x has to be an integer, right? The problem says "possible integer values of x." So, x must be an integer greater than 196 and less than or equal to 225.Now, how many integers are there in this range? Let's calculate that.The smallest integer greater than 196 is 197, and the largest integer less than or equal to 225 is 225 itself. So, the integers go from 197 up to 225.To find the number of integers in this range, I can subtract 196 from both ends. So, 197 - 196 = 1, and 225 - 196 = 29. So, the number of integers is 29.Wait, hold on, let me make sure. Sometimes when counting inclusive ranges, it's easy to make an off-by-one error. So, the number of integers from a to b inclusive is b - a + 1.In this case, a is 197 and b is 225. So, 225 - 197 + 1. Let me compute that.225 - 197 is 28, and 28 + 1 is 29. So, yes, that's correct. There are 29 integers from 197 to 225 inclusive.But let me double-check my initial inequality. I had 14 < √x ≤ 15, which led to 196 < x ≤ 225. So, x must be greater than 196 but less than or equal to 225.Therefore, the integer values of x start at 197 and go up to 225. So, 197, 198, ..., 225. As calculated, that's 29 numbers.Wait, just to make sure, let me test with some numbers. If x is 197, then √197 is approximately 14.035, and the ceiling of that is 15. Similarly, if x is 225, √225 is exactly 15, so the ceiling is 15. What about x = 196? √196 is 14, so the ceiling would be 14, which is not 15. So, 196 is excluded, which is why we start at 197.Similarly, if x is 226, √226 is approximately 15.033, so the ceiling would be 16, which is too big. So, 226 is excluded, which is why we stop at 225.Therefore, the number of integer values is 29.I think that's solid. I don't see any mistakes in my reasoning. So, I feel confident that the answer is 29.**Final Answer**The number of possible integer values of ( x ) is boxed{29}.