answer:We want to find the value of X in the given situation. We are given that the bee starts at point P_0 and flies 1 inch due east to point P_1. From there, for j ge X, the bee turns 30^{circ} counterclockwise and flies j+1 inches straight to point P_{j+1}. To find the distance from P_0 to P_{2015}, we can break down the bee's movements into segments and calculate the total distance. The bee's movements can be divided into two types of segments: 1. Straight segments: These are the segments where the bee flies straight from one point to another. 2. Turning segments: These are the segments where the bee turns 30^{circ} counterclockwise and changes direction. Let's analyze the straight segments first: For j ge X, the bee flies j+1 inches straight from P_j to P_{j+1}. So the length of the j-th straight segment is j+1 inches. Now let's analyze the turning segments: The bee turns 30^{circ} counterclockwise at P_j for j ge X. Since the bee turns 30^{circ} counterclockwise at each point, the total angle turned after n turns is 30^{circ} times n. To find the distance from P_0 to P_{2015}, we need to calculate the sum of the lengths of all the straight segments and add it to the distance covered during the turning segments. Let's calculate the sum of the lengths of the straight segments: The sum of the lengths of the straight segments can be written as: (X+1) + (X+2) + (X+3) + ... + 2015 Using the formula for the sum of an arithmetic series, we can simplify this expression: left(frac{(X+1) + 2015}{2}right) times (2015 - (X+1) + 1) Now, let's calculate the distance covered during the turning segments: The total angle turned after n turns is 30^{circ} times n. Since the bee reaches P_{2015}, the total number of turns is 2015 - X. So the total angle turned is 30^{circ} times (2015 - X). To find the distance covered during the turning segments, we can use the formula for the circumference of a circle: Distance = 2pi times r times (text{angle turned} / 360^{circ}) Since the bee starts at P_0 and turns counterclockwise, the distance covered during the turning segments is equal to the circumference of a circle with radius equal to the total distance covered by the bee during the straight segments. Now, we can set up the equation to find the value of X: Distance = (X+1) + (X+2) + (X+3) + ... + 2015 + 2pi times left(frac{(X+1) + 2015}{2}right) times (2015 - (X+1) + 1) Given that the distance is 1008sqrt{6}+1008sqrt{2}, we can substitute this value into the equation: 1008sqrt{6}+1008sqrt{2} = (X+1) + (X+2) + (X+3) + ... + 2015 + 2pi times left(frac{(X+1) + 2015}{2}right) times (2015 - (X+1) + 1) Simplifying this equation will give us the value of X. The answer is: 1

answer:Robi contributed 4000 to start the business. Rudy contributed 1/4 more money than Robi, which is 1/4 * 4000 = 1000 more. So Rudy contributed 4000 + 1000 = 5000. The total amount of money they contributed is 4000 + 5000 = 9000. They made a profit of 20 percent of the total amount, which is 20/100 * 9000 = 1800. Since they decided to share the profits equally, each of them received 1800 / 2 = 900. Therefore, each of them got 900. 900 The answer is: 900

answer:We want to rationalize the denominator of frac{5}{sqrt{125}}. To do this, we multiply the numerator and denominator by sqrt{125}. frac{5}{sqrt{125}}cdot frac{sqrt{125}}{sqrt{125}} = frac{5sqrt{125}}{sqrt{125}cdot sqrt{125}} = frac{5sqrt{125}}{sqrt{125^2}} = frac{5sqrt{125}}{125} Since sqrt{125} = sqrt{5^2cdot 5} = 5sqrt{5}, we can simplify this expression to frac{5cdot 5sqrt{5}}{125} = frac{25sqrt{5}}{125} = boxed{frac{sqrt{5}}{5}}. The answer is: frac{sqrt{5}}{5}

answer:We can use the formula d = rt, where d is the distance, r is the rate or speed, and t is the time. The distance from Carville to Nikpath is the same regardless of the speed, so we can set up the equation 70(4frac{1}{2}) = 60t. Simplifying, we have 315 = 60t. Dividing both sides by 60, we find t approx boxed{5.25}. The answer is: 5.25