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question:is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the x and y axes and so that the medians to the midpoints of the legs lie on the lines y = 3x + 1 and y = mx + X. The number of different constants m for which such a triangle exists is 2. What is the value of unknown variable X?
answer:To solve this problem, we need to determine the value of x, which represents the constant in the equation y = mx + X. We know that there are 2 different constants, m and X, for which such a triangle exists. The equation of the median to the midpoint of the leg parallel to the x-axis is y = 3x + 1. The equation of the median to the midpoint of the leg parallel to the y-axis is y = mx + X. For the medians to intersect, the two equations must have a solution. Let's set up the system of equations: 3x + 1 = mx + X To find the value of x, we need to find the values of m and X that make the system of equations consistent. Since there are 2 different constants for which such a triangle exists, we can conclude that the system of equations has infinitely many solutions. Therefore, there is no unique value for the unknown variable x in this problem. The answer is: 2
question:Jenna and her four friends are at a fair and want to ride the roller coaster. However, only three people can fit in each car. How many different groups of three can be formed from the five of them?
answer:We can choose 3 people out of the 5 to form a group, which can be done in binom{5}{3} = boxed{10} ways. The answer is: 10
question:What is the closest integer to the cube root of 100?
answer:We can estimate that 5^3=125 is slightly larger than 100, while 4^3=64 is slightly smaller than 100. Therefore, the cube root of 100 is slightly larger than 4. The closest integer to the cube root of 100 is boxed{5}.The answer is: 5
question:Facebook decided to award a productivity bonus to all its female employees who are mothers. This productivity bonus will total 25% of Facebook's annual earnings, which was 5,000,000 for the year 2020. It is known that Facebook employs 3300 employees; one-third are men, and of the women, 1200 are not mothers. How much was the bonus that each female mother employee received, assuming each one received an equal amount?
answer:One-third of Facebook's employees are men, so 3300/3 = 1100 employees are men. This means that 3300 - 1100 = 2200 employees are women. Out of the women, 1200 are not mothers, so there are 2200 - 1200 = 1000 female mother employees. The productivity bonus is 25% of Facebook's annual earnings, so it is 25/100 * 5,000,000 = 1,250,000. If each female mother employee received an equal amount, then each one received 1,250,000 / 1000 = 1250. Therefore, each female mother employee received a bonus of 1250. 1250 The answer is: 1250