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question:A bowl was filled with 10 ounces of water, and some amount of water evaporated each day during a 50day period. 6% of the original amount of water evaporated during this period. How many ounces of water evaporated each day?
answer:If 6% of the original amount of water evaporated during the 50-day period, we first need to find out how much water that is in ounces. 6% of 10 ounces is: (6/100) * 10 = 0.6 ounces So, 0.6 ounces of water evaporated over the 50-day period. To find out how much water evaporated each day, we divide the total amount of evaporated water by the number of days: 0.6 ounces / 50 days = 0.012 ounces per day Therefore, boxed{0.012} ounces of water evaporated each day.
question:Convert the number 2202_5 to a base 10 integer.
answer:The number given in base 5 is 2202_5. To convert 2202_5 into base 10, each digit must be multiplied by 5 raised to the power of its position, counting from right to left starting with 0. [ 2202_5 = 2 cdot 5^3 + 2 cdot 5^2 + 0 cdot 5^1 + 2 cdot 5^0 ] Computing each term: - 2 cdot 5^3 = 2 cdot 125 = 250 - 2 cdot 5^2 = 2 cdot 25 = 50 - 0 cdot 5^1 = 0 cdot 5 = 0 - 2 cdot 5^0 = 2 cdot 1 = 2 Adding them together gives the base 10 number: [ 250 + 50 + 0 + 2 = boxed{302} ]
question:On a balance scale, 4 green balls balance 9 blue balls, 3 yellow balls balance 7 blue balls, and 9 blue balls balance 6 white balls. Determine how many blue balls are necessary to balance 5 green, 4 yellow, and 3 white balls.
answer:First, denote weights of green, yellow, white, and blue balls as G, Y, W, and B, respectively. Given: - 4G = 9B implies G = frac{9}{4}B - 3Y = 7B implies Y = frac{7}{3}B - 9B = 6W implies W = frac{3}{2}B We need to calculate the weight of 5 green, 4 yellow, and 3 white balls in terms of blue balls. Using the relationships: - The total weight of 5 green balls is 5G = 5 left(frac{9}{4}Bright) = frac{45}{4}B - The total weight of 4 yellow balls is 4Y = 4 left(frac{7}{3}Bright) = frac{28}{3}B - The total weight of 3 white balls is 3W = 3 left(frac{3}{2}Bright) = frac{9}{2}B Adding these weights together: [ frac{45}{4}B + frac{28}{3}B + frac{9}{2}B = left(frac{45}{4} + frac{28}{3} + frac{9}{2}right)B = left(frac{135}{12} + frac{112}{12} + frac{54}{12}right)B = frac{301}{12}B ] Thus, boxed{frac{301}{12}} blue balls are needed to balance 5 green, 4 yellow, and 3 white balls.
question:The parabolas defined by the equations y = 3x^2 6x + 3 and y = x^2 3x + 3 intersect at points (p,q) and (r,s), where r ge p. What is r p?
answer:Set the equations equal for intersection points: 3x^2 - 6x + 3 = -x^2 - 3x + 3. Combine like terms to find: 3x^2 - 6x + 3 + x^2 + 3x - 3 = 0 Rightarrow 4x^2 - 3x = 0. Factor out the common terms: x(4x - 3) = 0. Set each factor equal to zero: 1. x = 0 2. 4x - 3 = 0 Rightarrow x = frac{3}{4}. Hence, the x-coordinates of the intersection points are 0 and frac{3}{4}. Thus, p = 0 and r = frac{3}{4}, leading to: r - p = frac{3}{4} - 0 = boxed{frac{3}{4}}.