💡Okay, so I have this problem here about finding the standard equation of a parabola with a given focus. The focus is at F(0,1), and there are four options to choose from: A is x² = 4y, B is x² = 2y, C is y² = 4x, and D is y² = 2x. I need to figure out which one is the correct equation.First, I remember that the standard form of a parabola depends on whether it opens up, down, left, or right. Since the focus is at (0,1), which is on the y-axis, I think the parabola must open either upward or downward. If the focus is above the vertex, it opens upward, and if it's below, it opens downward.Looking at the coordinates, the focus is at (0,1). If I assume the vertex is at the origin (0,0), which is a common case for standard equations, then the focus being at (0,1) means it's one unit above the vertex. So, the parabola opens upward.Now, I recall that the standard form of a parabola that opens upward or downward is x² = 4py, where p is the distance from the vertex to the focus. In this case, since the focus is at (0,1) and the vertex is at (0,0), p should be 1.So, plugging p = 1 into the standard form, I get x² = 4*1*y, which simplifies to x² = 4y. That matches option A.Wait, but let me double-check. If p is the distance from the vertex to the focus, then in the equation x² = 4py, p is positive if it opens upward and negative if it opens downward. Since our focus is above the vertex, p is positive, so everything checks out.Just to be thorough, let's consider the other options. Option B is x² = 2y. That would mean 4p = 2, so p = 0.5. But our focus is at (0,1), which is p = 1, not 0.5. So B doesn't seem right.Options C and D are y² = 4x and y² = 2x, respectively. These are equations of parabolas that open to the right or left. Since our focus is on the y-axis, the parabola should open upward or downward, not sideways. So C and D can be ruled out.Therefore, option A, x² = 4y, is the correct equation for the parabola with focus at (0,1).

💡Okay, so I need to find the 150th digit after the decimal point in the decimal representation of 1/13. Hmm, I remember that when you divide 1 by a number like 13, you get a repeating decimal. Let me try to recall what that looks like.First, I think I should actually perform the division of 1 by 13 to see what the decimal expansion is. So, let's see, 13 goes into 1 zero times, then we add a decimal point and some zeros to continue the division. 13 into 10 is 0, so we write 0. and then bring down a zero to make it 10. 13 into 10 is still 0, so we write another 0. Now we have 100. 13 into 100 is 7 because 13 times 7 is 91. Subtracting 91 from 100 gives us 9. Bring down another zero to make it 90. 13 into 90 is 6 because 13 times 6 is 78. Subtracting 78 from 90 gives us 12. Bring down another zero to make it 120. 13 into 120 is 9 because 13 times 9 is 117. Subtracting 117 from 120 gives us 3. Bring down another zero to make it 30. 13 into 30 is 2 because 13 times 2 is 26. Subtracting 26 from 30 gives us 4. Bring down another zero to make it 40. 13 into 40 is 3 because 13 times 3 is 39. Subtracting 39 from 40 gives us 1. Now we're back to where we started with 10, so the pattern will repeat.So, the decimal expansion of 1/13 is 0.076923076923..., repeating every 6 digits. That means the sequence "076923" repeats indefinitely.Now, to find the 150th digit after the decimal point, I need to figure out where 150 falls in terms of these repeating cycles. Since the cycle length is 6, I can divide 150 by 6 to see how many complete cycles there are and what the remainder is.Let's do that division: 150 divided by 6. 6 times 25 is 150, so there's no remainder. That means the 150th digit is the last digit of the 25th cycle.Looking back at the repeating sequence "076923", the last digit is 3. Therefore, the 150th digit after the decimal point in the decimal representation of 1/13 is 3.

💡First, I need to determine the number of child workers employed by the contractor. I'll start by calculating the total daily wages paid to the male and female workers.For the male workers:- There are 20 male workers.- Each male worker is paid Rs. 25 per day.- Total daily wage for male workers = 20 * 25 = Rs. 500.For the female workers:- There are 15 female workers.- Each female worker is paid Rs. 20 per day.- Total daily wage for female workers = 15 * 20 = Rs. 300.Let ( C ) represent the number of child workers. Each child worker is paid Rs. 8 per day, so the total daily wage for child workers is ( 8C ).The total number of workers is the sum of male, female, and child workers:- Total workers = 20 + 15 + ( C ) = 35 + ( C ).The total daily wage paid to all workers is:- Total wage = 500 + 300 + ( 8C ) = 800 + ( 8C ).The average wage per day is given as Rs. 21. I can set up the equation:[ frac{800 + 8C}{35 + C} = 21 ]To solve for ( C ), I'll multiply both sides by ( 35 + C ):[ 800 + 8C = 21(35 + C) ][ 800 + 8C = 735 + 21C ]Next, I'll move all terms involving ( C ) to one side and constants to the other:[ 800 - 735 = 21C - 8C ][ 65 = 13C ]Finally, I'll solve for ( C ):[ C = frac{65}{13} ][ C = 5 ]Therefore, the contractor employs 5 child workers.

💡Okay, so I have this problem with two parts. Let me try to figure it out step by step. First, part (1) asks for the general equation of line l and the rectangular coordinate equation of curve C. Starting with line l, I know it's given in parametric form:[begin{cases}x = frac{1}{2}t y = frac{sqrt{3}}{2}t - 1end{cases}]where ( t ) is the parameter. I need to eliminate the parameter ( t ) to find the general equation. From the first equation, ( x = frac{1}{2}t ), I can solve for ( t ):[t = 2x]Now, substitute this into the second equation:[y = frac{sqrt{3}}{2}(2x) - 1 = sqrt{3}x - 1]So, the general equation of line l is ( y = sqrt{3}x - 1 ). That seems straightforward.Next, for curve C, the polar equation is given as:[rho^2 - 2rho sintheta - 3 = 0]I need to convert this into rectangular coordinates. I remember that in polar coordinates, ( rho^2 = x^2 + y^2 ) and ( rho sintheta = y ). Let me substitute these into the equation:[x^2 + y^2 - 2y - 3 = 0]Hmm, this looks like the equation of a circle. Let me rearrange it to the standard form. I'll complete the square for the y-terms:[x^2 + (y^2 - 2y) = 3]To complete the square for ( y^2 - 2y ), I take half of -2, which is -1, square it to get 1, and add it to both sides:[x^2 + (y^2 - 2y + 1) = 3 + 1]So,[x^2 + (y - 1)^2 = 4]This is indeed a circle with center at (0, 1) and radius 2. Okay, so the rectangular equation for curve C is ( x^2 + (y - 1)^2 = 4 ).Alright, part (1) seems done. Now, part (2) asks for the length of the line segment intercepted by line l on curve C. That means I need to find where line l intersects curve C and then find the distance between those two intersection points.First, let me write down the equations again:- Line l: ( y = sqrt{3}x - 1 )- Curve C: ( x^2 + (y - 1)^2 = 4 )I can substitute the expression for ( y ) from line l into the equation of curve C. Let's do that:[x^2 + (sqrt{3}x - 1 - 1)^2 = 4]Simplify inside the parentheses:[x^2 + (sqrt{3}x - 2)^2 = 4]Now, expand ( (sqrt{3}x - 2)^2 ):[(sqrt{3}x)^2 - 2 cdot sqrt{3}x cdot 2 + 2^2 = 3x^2 - 4sqrt{3}x + 4]So, substituting back:[x^2 + 3x^2 - 4sqrt{3}x + 4 = 4]Combine like terms:[4x^2 - 4sqrt{3}x + 4 = 4]Subtract 4 from both sides:[4x^2 - 4sqrt{3}x = 0]Factor out 4x:[4x(x - sqrt{3}) = 0]So, the solutions are:[4x = 0 Rightarrow x = 0]and[x - sqrt{3} = 0 Rightarrow x = sqrt{3}]Now, find the corresponding y-values using the equation of line l, ( y = sqrt{3}x - 1 ).For ( x = 0 ):[y = sqrt{3}(0) - 1 = -1]So, one point is (0, -1).For ( x = sqrt{3} ):[y = sqrt{3}(sqrt{3}) - 1 = 3 - 1 = 2]So, the other point is (( sqrt{3} ), 2).Now, I need to find the distance between these two points: (0, -1) and (( sqrt{3} ), 2). The distance formula is:[sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}]Plugging in the values:[sqrt{(sqrt{3} - 0)^2 + (2 - (-1))^2} = sqrt{(sqrt{3})^2 + (3)^2} = sqrt{3 + 9} = sqrt{12} = 2sqrt{3}]So, the length of the line segment intercepted by line l on curve C is ( 2sqrt{3} ).Wait, let me double-check my steps to make sure I didn't make a mistake. Starting from substituting ( y = sqrt{3}x - 1 ) into the circle equation, expanding, simplifying, factoring—all seems correct. The solutions for x are 0 and ( sqrt{3} ), which give y as -1 and 2, respectively. The distance between these points is indeed ( 2sqrt{3} ). I think that's right.Another way to think about it is that since the line passes through the center of the circle (0,1), the intercepted segment is a diameter. But wait, the radius is 2, so the diameter would be 4. But according to my calculation, it's ( 2sqrt{3} ). Hmm, that doesn't match. Maybe my assumption is wrong.Wait, let me see. The center of the circle is (0,1), and the line passes through (0,-1) and (( sqrt{3} ), 2). Does it pass through the center? Let me check if (0,1) is on the line. Plugging x=0 into ( y = sqrt{3}x - 1 ), we get y = -1, which is not 1. So, the line does not pass through the center. Therefore, the intercepted segment is not a diameter. So, my initial calculation of ( 2sqrt{3} ) is correct.Alternatively, I can use the formula for the length of a chord in a circle: ( 2sqrt{r^2 - d^2} ), where ( r ) is the radius and ( d ) is the distance from the center to the line. Let me try that.First, find the distance from the center (0,1) to the line ( y = sqrt{3}x - 1 ). The formula for the distance from a point ( (x_0, y_0) ) to the line ( Ax + By + C = 0 ) is:[d = frac{|Ax_0 + By_0 + C|}{sqrt{A^2 + B^2}}]First, rewrite the line equation in standard form:[y = sqrt{3}x - 1 Rightarrow sqrt{3}x - y - 1 = 0]So, A = ( sqrt{3} ), B = -1, C = -1.Plugging in the center (0,1):[d = frac{|sqrt{3}(0) + (-1)(1) - 1|}{sqrt{(sqrt{3})^2 + (-1)^2}} = frac{|0 - 1 - 1|}{sqrt{3 + 1}} = frac{|-2|}{2} = frac{2}{2} = 1]So, the distance from the center to the line is 1. The radius is 2, so the length of the chord is:[2sqrt{r^2 - d^2} = 2sqrt{2^2 - 1^2} = 2sqrt{4 - 1} = 2sqrt{3}]This matches my previous result. So, I was correct. The length is ( 2sqrt{3} ).I think I confused myself earlier by thinking the line passes through the center, but it doesn't. The distance from the center to the line is 1, which is less than the radius, so the line intersects the circle at two points, and the chord length is ( 2sqrt{3} ).So, to summarize:1. The general equation of line l is ( y = sqrt{3}x - 1 ).2. The rectangular equation of curve C is ( x^2 + (y - 1)^2 = 4 ).3. The length of the intercepted line segment is ( 2sqrt{3} ).I think that's all. I didn't make any calculation errors this time, I believe.**Final Answer**(1) The general equation of line ( l ) is ( y = sqrt{3}x - 1 ) and the rectangular coordinate equation of curve ( C ) is ( x^2 + (y - 1)^2 = 4 ).(2) The length of the line segment intercepted by line ( l ) on curve ( C ) is (boxed{2sqrt{3}}).

💡Okay, so I've got this problem here: the function f(x) is defined as x raised to the power of (2 - m), and it's defined on the interval [-3 - m, m² - m]. They also tell me that this function is an odd function, and I need to find f(m). Hmm, okay, let me try to unpack this step by step.First, let me recall what an odd function is. An odd function satisfies the condition f(-x) = -f(x) for all x in its domain. So, that means if I plug in -x into the function, it should be the same as taking the negative of the function evaluated at x. That's the key property here.Now, the function given is f(x) = x^(2 - m). So, for this function to be odd, it needs to satisfy f(-x) = -f(x). Let me write that out:f(-x) = (-x)^(2 - m) -f(x) = -x^(2 - m)So, for these two expressions to be equal for all x in the domain, the exponents must satisfy certain conditions. Let me think about the exponent (2 - m). If I raise -x to that exponent, I need to get the negative of x raised to that exponent. So, (-x)^(2 - m) = -x^(2 - m). Hmm, when does that happen?Well, let's consider the exponent (2 - m). If (2 - m) is an odd integer, then (-x)^(2 - m) would be equal to -x^(2 - m) because (-x)^odd = -x^odd. On the other hand, if (2 - m) is an even integer, then (-x)^(2 - m) would be equal to x^(2 - m), which is not equal to -x^(2 - m). So, for the function to be odd, the exponent (2 - m) must be an odd integer. That gives me the condition that (2 - m) is odd.So, 2 - m must be odd. Let me write that as:2 - m = 2k + 1, where k is an integer.Solving for m:m = 2 - (2k + 1) m = 1 - 2kSo, m must be an odd integer. That's one condition.Now, moving on to the domain of the function. The interval is given as [-3 - m, m² - m]. Since the function is defined on this interval, the interval must be valid, meaning the left endpoint must be less than or equal to the right endpoint. So:-3 - m ≤ m² - mLet me simplify this inequality:-3 - m ≤ m² - m Add m to both sides: -3 ≤ m²Hmm, m² is always non-negative, so -3 ≤ m² is always true. So, this condition doesn't give me any new information. Maybe I need to consider something else about the domain.Wait, since the function is odd, it's symmetric about the origin. That means the domain should also be symmetric about zero. So, the interval [-a, a] for some a. Let me check if the given interval is symmetric.Given interval: [-3 - m, m² - m]For it to be symmetric about zero, the left endpoint should be the negative of the right endpoint. So:- (m² - m) = -3 - mLet me write that equation:- (m² - m) = -3 - m Multiply out the left side: - m² + m = -3 - mBring all terms to one side:- m² + m + 3 + m = 0 Combine like terms: - m² + 2m + 3 = 0 Multiply both sides by -1 to make it standard: m² - 2m - 3 = 0Now, I have a quadratic equation: m² - 2m - 3 = 0. Let me solve for m.Using the quadratic formula:m = [2 ± sqrt(4 + 12)] / 2 m = [2 ± sqrt(16)] / 2 m = [2 ± 4] / 2So, two solutions:m = (2 + 4)/2 = 6/2 = 3 m = (2 - 4)/2 = (-2)/2 = -1So, m can be either 3 or -1. Now, earlier I found that m must be an odd integer because (2 - m) must be odd. Let's check if these values satisfy that.For m = 3:2 - m = 2 - 3 = -1, which is odd. Good.For m = -1:2 - m = 2 - (-1) = 3, which is also odd. Good.So both m = 3 and m = -1 satisfy the condition that (2 - m) is odd, which is necessary for the function to be odd.Now, let's check if these values of m make sense in the context of the function's domain.First, m = 3:The interval becomes [-3 - 3, 3² - 3] = [-6, 9 - 3] = [-6, 6]. So, the interval is [-6, 6], which is symmetric about zero, as expected for an odd function. The function is f(x) = x^(2 - 3) = x^(-1) = 1/x. Hmm, but wait, 1/x is undefined at x = 0. So, does that mean the function isn't defined at x = 0? But the interval includes 0 because it's from -6 to 6. So, is this a problem?Well, the function f(x) = 1/x is indeed undefined at x = 0, but the interval [-6, 6] includes 0. So, does that mean m = 3 is invalid? Because the function isn't defined at x = 0, which is within the interval. Hmm, maybe m = 3 is not acceptable because the function isn't defined on the entire interval. So, perhaps m = 3 is not a valid solution.Now, let's check m = -1:The interval becomes [-3 - (-1), (-1)^2 - (-1)] = [-3 + 1, 1 + 1] = [-2, 2]. So, the interval is [-2, 2], which is symmetric about zero. The function is f(x) = x^(2 - (-1)) = x^(3). So, f(x) = x^3. That's a standard odd function, and it's defined for all real numbers, including the interval [-2, 2]. So, m = -1 seems to be a valid solution.Therefore, m = -1 is the valid solution, and m = 3 is invalid because the function isn't defined at x = 0 in that case.So, now, the question is to find f(m). Since m = -1, we need to compute f(-1).Given f(x) = x^3, so f(-1) = (-1)^3 = -1.Therefore, f(m) = -1.Wait, let me double-check everything to make sure I didn't make a mistake.First, confirming that m = -1 gives an odd function:f(x) = x^3, which is indeed odd because f(-x) = (-x)^3 = -x^3 = -f(x).And the interval is [-2, 2], which is symmetric about zero, so that's good.For m = 3, f(x) = x^(-1) = 1/x, which is odd, but it's undefined at x = 0, which is within the interval [-6, 6]. So, the function isn't defined on the entire interval, which might make m = 3 invalid. So, m = -1 is the only valid solution.Therefore, f(m) = f(-1) = (-1)^3 = -1.Yeah, that seems correct.

💡First, I need to calculate the discounts Todd received on the candy bar and the hamburger. The candy bar costs 1.14, and with a 12% discount, the discount amount is 0.1368. Subtracting this from the original price gives the discounted candy bar price of 1.0032. Similarly, the hamburger costs 3.99, and a 12% discount amounts to 0.4788. Subtracting this discount from the original price results in a discounted hamburger price of 3.5112.Next, I'll calculate the discount on the box of cookies. The cookies cost 2.39, and with a 15% discount, the discount amount is 0.3585. Subtracting this from the original price gives the discounted cookie price of 2.0315.Now, I'll add up the costs of all the items after applying the discounts:- Candy bar: 1.0032- Cookies: 2.0315- Soda: 1.75- Chips: 1.85- Juice: 2.69- Hamburger: 3.5112Adding these together, the total cost after discounts is 12.8369.Finally, I'll calculate the sales tax on this total amount. An 8.5% sales tax on 12.8369 amounts to 1.0911. Adding this tax to the total cost after discounts gives the final amount Todd spent: 13.928. Rounding this to two decimal places, Todd spent a total of 13.93.