MIT

🔍

question:The mean of the set of numbers {87,85,80,83,84,x} is X. What is the median of the set of six numbers? Express your answer as a decimal to the nearest tenth. If we know the answer to the above question is 83.5, what is the value of unknown variable X?

answer:We are given the set of numbers {87,85,80,83,84,x} and the mean of this set is X. To find the median of the set of six numbers, we first need to arrange the numbers in ascending order: {80,83,84,85,87,x}. The median is the middle number of the set when arranged in ascending order. Since we have an even number of numbers, the median is the average of the two middle numbers. In this case, the two middle numbers are 84 and 85. Therefore, the median is (84 + 85) / 2 = 169 / 2 = 84.5. Given that the median is 83.5, we can set up the equation: 83.5 = (84 + 85) / 2 Multiplying both sides of the equation by 2, we get: 167 = 84 + 85 Simplifying, we have: 167 = 169 However, this equation is not correct. Therefore, there is no solution for the value of X that would give a median of 83.5 for the given set of numbers. The answer is: 83.5

question:Anthony and his friend Leonel read about the importance of keeping pets at home and decided to start adopting cats and dogs from the local rescue center. Anthony has 12 cats and dogs, 2/3 of which are cats. Leonel has half times as many cats as Anthony and seven more dogs than Anthony. How many animals in total do the two have?

answer:Anthony has 2/3 of 12 animals as cats, so he has 2/3 x 12 = 8 cats. The remaining animals are dogs, so Anthony has 12 - 8 = 4 dogs. Leonel has half times as many cats as Anthony, so he has 1/2 x 8 = 4 cats. Leonel also has seven more dogs than Anthony, so he has 4 + 7 = 11 dogs. Therefore, Anthony and Leonel have a total of 8 cats + 4 cats + 4 dogs + 11 dogs = 27 animals. 27 The answer is: 27

question:A cube has a surface area of 216 square centimeters. What is the volume of the cube, in cubic centimeters?

answer:A cube has 6 faces, each with side length s. The surface area of a cube is given by 6s^2. We know that 6s^2=216, so s^2=36. Taking the square root of both sides, we find that s=6. The volume of a cube is given by s^3, so the volume of this cube is 6^3=boxed{216} cubic centimeters. The answer is: 216

question:What is the unique value of b for the smallest positive real number a where there exists a positive real number b such that all the roots of the polynomial x^3 - ax^2 + bx - a are real?

answer:By Vieta's formulas, the sum of the roots of the polynomial is equal to a, which means that the roots are positive since a is positive. Let r_1, r_2, and r_3 be the roots of the polynomial. By Vieta's formulas, we have [r_1 + r_2 + r_3 = a,] [r_1r_2 + r_2r_3 + r_3r_1 = b,] [r_1r_2r_3 = a.] Since all the roots are positive, we have r_1r_2 > 0, r_2r_3 > 0, and r_3r_1 > 0. Hence, b = r_1r_2 + r_2r_3 + r_3r_1 > 0. The smallest positive real number a occurs when the roots are equal, so r_1 = r_2 = r_3. Substituting this into the equations, we have [3r_1 = a,] [3r_1^2 = b,] [r_1^3 = a.] Substituting a = 3r_1 into the last equation, we have r_1^3 = 3r_1. Dividing both sides by r_1, we have r_1^2 = 3. Taking the positive square root, we have r_1 = sqrt{3}. Substituting this into the equations, we have a = 3sqrt{3} and b = 3(sqrt{3})^2 = 9. Hence, the unique value of b is boxed{9}. The answer is: 9