answer:First, we find the coordinates of point M. Since M is on the x-axis, its y-coordinate must be 0. By plugging y = 0 into the equation of line l, we can solve for M's x-coordinate: begin{align*} 2x - y - 4 &= 0 2x - 0 - 4 &= 0 2x &= 4 x &= 2. end{align*} Therefore, the coordinates of point M are (2, 0). Next, we determine the slope of line l. The slope is the coefficient of x when the equation is in the slope-intercept form (y = mx + b). To find it, we solve for y: begin{align*} 2x - y - 4 &= 0 y &= 2x - 4. end{align*} The slope of l is 2. Since the line we are looking for is perpendicular to line l, its slope will be the negative reciprocal of 2. Thus, the slope of the new line m' is -frac{1}{2}. With the slope of the new line and the coordinates of point M, we can apply the point-slope form of the equation of a line (y - y_1 = m(x - x_1)): begin{align*} y - 0 &= -frac{1}{2}(x - 2) 2y &= -(x - 2) 2y &= -x + 2 x + 2y - 2 &= 0. end{align*} The equation of the line that passes through M and is perpendicular to line l is boxed{x + 2y - 2 = 0}.

answer:1. Let the length of the parallelepiped be denoted by ( n ) cm. According to the problem, the width is ( n - 2 ) cm and the height is ( n - 4 ) cm. 2. The total number of 1 cm³ cubes that can be obtained by cutting the parallelepiped is: [ n times (n - 2) times (n - 4) ] 3. Pieces having any painted face can be calculated by subtracting the unpainted cubic pieces from the total number of cubic pieces. The unpainted cubic pieces would be those pieces that are entirely inside the parallelepiped and not on the surface, meaning they would have 1 cm removed from each dimension: [ (n - 2) times (n - 4) times (n - 6) ] 4. According to the problem, one-third of the cubes have at least one red face, and the rest, which would be two-thirds of the cubes, have no red faces. Therefore, we set up the following equation to reflect this distribution: [ (n - 2)(n - 4)(n - 6) = frac{2}{3} n (n - 2)(n - 4) ] 5. First, note that ( n geq 7 ) since the dimensions must be positive integers. Simplifying the equation: [ (n - 6) = frac{2}{3} n implies 3(n - 6) = 2n implies 3n - 18 = 2n implies n = 18 ] 6. Checking this value confirms that it satisfies the equation and falls within the allowable range of values for ( n ). 7. Verifying the dimensions: - Length ( n = 18 ) cm - Width ( n - 2 = 16 ) cm - Height ( n - 4 = 14 ) cm Conclusion: [ boxed{18 text{ cm}} ]

answer:To solve this, we need to determine all the elements in (G) that are squares, given the group relations. Let's go through the problem step by step. 1. **Group Relations Interpretation**: - ( G ) is generated by ( g ) and ( h ). - The given relations are: - ( g^4 = 1 ) - ( g^2 ne 1 ) - ( h^7 = 1 ) - ( h ne 1 ) - ( ghg^{-1}h = 1 ) 2. **Implications of Relations**: - From ( ghg^{-1}h = 1 ), we can rearrange and solve for ( ghg^{-1} ): [ ghg^{-1}h = 1 implies ghg^{-1} = h^{-1} ] - Since ( h^{-1} = h^6 ) (as ( h^6 cdot h = h^7 = 1 )), we have: [ ghg^{-1} = h^6 ] 3. **Expressing Elements in Standard Form**: - Any element in ( G ) can be written as ( h^ng^m ) where ( n ) ranges from 0 to 6 (by ( h^7 = 1 )), and ( m ) ranges from 0 to 3 (by ( g^4 = 1 )). - Using the relation ( gh = h^6g ), we get: [ gh = h^6g ] 4. **Calculating Squares**: - Let's examine different forms: - ( (h^n g)^2 ): [ (h^n g)^2 = h^n g h^n g ] Using ( gh = h^6 g ): [ h^n g h^n g = h^n (gh^n) g = h^n (h^6 g h^{n-6}) g = h^6 g^2 quad (text{since } h^{n-6} = h^{-(7-n)} text{ and this cycle repeats every 7 steps}) ] Simplifying further: [ (h^n g)^2 = h^{n + n} g^2 = h^{2n} g^2 ] - ( (h^n g^2)^2 ): [ (h^n g^2)^2 = h^n g^2 h^n g^2 = h^n h^n g^4 = h^{2n} ] - ( (h^n g^3)^2 ): [ (h^n g^3)^2 = h^n g^3 h^n g^3 = h^n (gh^{-1}) g^3 = h^6 g^2 ] 5. **List of Squares**: - From these calculations: - (1 = (h^0)^2) - (g^2 = (h^n g)^2) - (h = (h^4)^2) - (h^2 = (h^2)^2) - (h^3 = (h^6h^4)^2) - (h^4 = (h^4 h^4)^2) - (h^5 = (h h^4)^2) - (h^6 = (h^2 h^4)^2) 6. **Verifying All Squares Are Distinct**: - All elements: (1, g^2, h, h^2, h^3, h^4, h^5, h^6) are distinct. - If any power of ( h ) equaled another, ( h ) would have to be equal to 1 to maintain equality under given constraints, which contradicts ( h ne 1 ). Conclusion: The elements of ( G ) which are squares are: [ boxed{1, g^2, h, h^2, h^3, h^4, h^5, h^6} ]

answer:1. **Convert the swath width and overlap to feet:** The swath width is 30 inches minus a 6 inch overlap, so the effective swath width is: [ 30 - 6 = 24 text{ inches} = frac{24}{12} text{ feet} = 2 text{ feet} ] 2. **Calculate the number of strips needed:** Moe mows parallel to the 120-foot side, so each strip is 120 feet long. The total width of the lawn is 180 feet. Since each effective swath is 2 feet wide, the number of strips required is: [ frac{180 text{ feet}}{2 text{ feet/strip}} = 90 text{ strips} ] 3. **Calculate the total distance Moe mows:** Each strip is 120 feet long and there are 90 strips, so the total distance Moe mows is: [ 90 text{ strips} times 120 text{ feet/strip} = 10800 text{ feet} ] 4. **Calculate the time required to mow the lawn:** Moe's mowing rate is 4000 feet per hour. Therefore, the time required to mow the lawn is: [ frac{10800 text{ feet}}{4000 text{ feet/hour}} = 2.7 text{ hours} ] 5. **Conclusion:** Since Moe only has 2 hours before he needs to leave and it will take him approximately 2.7 hours to mow the lawn, he will not be able to finish mowing the lawn in time. The answer is text{No}. The final answer is boxed{B}