Appearance
question:Jenny collects cans, bottles, and glass jars to take to the recycling center. Each bottle weighs 6 ounces, each can weighs 2 ounces, and each glass jar weighs 8 ounces. Jenny can carry a total of 100 ounces. She decides to collect 20 cans, as many glass jars as possible without exceeding the weight limit, and the rest of the weight with bottles. The recycling center pays 10 cents per bottle, 3 cents per can, and 12 cents per glass jar. How much money does Jenny make in total (in cents) from recycling these items, and what is the breakdown of cans, bottles, and glass jars she carries in this load?
answer:First, let's calculate the total weight of the 20 cans Jenny decides to collect: 20 cans * 2 ounces/can = 40 ounces Now, let's subtract the weight of the cans from the total weight Jenny can carry: 100 ounces - 40 ounces = 60 ounces remaining for glass jars and bottles. Next, let's calculate how many glass jars Jenny can carry without exceeding the weight limit. Since each glass jar weighs 8 ounces, we divide the remaining weight by the weight of one glass jar: 60 ounces / 8 ounces/glass jar = 7.5 glass jars Since Jenny can't carry half a glass jar, she can carry 7 glass jars. The weight of these glass jars is: 7 glass jars * 8 ounces/glass jar = 56 ounces Now, let's subtract the weight of the glass jars from the remaining weight to find out how much weight is left for bottles: 60 ounces - 56 ounces = 4 ounces Since each bottle weighs 6 ounces and Jenny only has 4 ounces of carrying capacity left, she can't carry any more bottles. So, the breakdown of items Jenny carries is: - 20 cans - 7 glass jars - 0 bottles Now, let's calculate how much money Jenny makes from each type of item: Cans: 20 cans * 3 cents/can = 60 cents Glass jars: 7 glass jars * 12 cents/glass jar = 84 cents Bottles: 0 bottles * 10 cents/bottle = 0 cents Finally, let's add up the money Jenny makes from all the items: 60 cents (from cans) + 84 cents (from glass jars) + 0 cents (from bottles) = 144 cents Jenny makes a total of boxed{144} cents from recycling these items.
question:Eight balls, numbered 3, 4, 5, 6, 7, 8, 9, 10, are placed in a hat. Each ball is equally likely to be chosen. If one ball is chosen, what is the probability that the number on the selected ball is a composite number?
answer:Of the numbers 3, 4, 5, 6, 7, 8, 9, 10: - Prime numbers: 3, 5, 7 - Composite numbers: 4, 6, 8, 9, 10 There are 5 composite numbers out of 8 total numbers. Thus, the probability of choosing a ball with a composite number is (frac{5}{8}). Conclusion with boxed answer: The probability of choosing a ball that has a composite number is boxed{frac{5}{8}}.
question:Vijay sells a cupboard at a certain percentage below cost price. Had he got Rs. 2086 more, he would have made a profit of 14%. The cost price of the cupboard is Rs. 7449.999999999999. What is the percentage below cost price at which he sold the cupboard?
answer:Let's denote the cost price of the cupboard as CP and the selling price as SP. We are given that CP is Rs. 7449.999999999999 (which we can round to Rs. 7450 for simplicity). We are also told that if Vijay had sold the cupboard for Rs. 2086 more, he would have made a 14% profit. This means that the selling price that would have given him a 14% profit is: SP (for 14% profit) = CP + (14% of CP) SP (for 14% profit) = Rs. 7450 + (0.14 × Rs. 7450) SP (for 14% profit) = Rs. 7450 + Rs. 1043 SP (for 14% profit) = Rs. 8493 Now, we know that the actual selling price (SP) was Rs. 2086 less than this amount. So, the actual selling price is: Actual SP = SP (for 14% profit) - Rs. 2086 Actual SP = Rs. 8493 - Rs. 2086 Actual SP = Rs. 6407 Now, we can calculate the percentage below the cost price at which Vijay sold the cupboard. The difference between the cost price and the selling price is: Difference = CP - Actual SP Difference = Rs. 7450 - Rs. 6407 Difference = Rs. 1043 The percentage below the cost price is then: Percentage below CP = (Difference / CP) × 100% Percentage below CP = (Rs. 1043 / Rs. 7450) × 100% Percentage below CP = (0.14) × 100% Percentage below CP = 14% Therefore, Vijay sold the cupboard at boxed{14%} below the cost price.
question:Given the parabola y^{2}=4x, its focus intersects the parabola at points A(x_{1},y_{1}) and B(x_{2},y_{2}). If |AB|=7, find the distance from the midpoint M of AB to the parabola's directrix ( quad ) A: dfrac {7}{2} B: dfrac {5}{2} C: 2 D: dfrac {9}{2}
answer:From the equation of the parabola y^{2}=4x, we have p=2. Thus, its focus is F(1,0), and the equation of the directrix is x=-1. According to the definition of a parabola, we have |AB|=7=|AF|+|BF|=(x_{1}+1)+(x_{2}+1), which implies x_{1}+x_{2}=5. The midpoint M of AB is given by M( dfrac {x_{1}+x_{2}}{2}, dfrac {y_{1}+y_{2}}{2}). The distance from M to the directrix is dfrac {x_{1}+x_{2}}{2}+1=dfrac {7}{2}. Thus, the correct answer is A, boxed{dfrac{7}{2}}. The focus of the parabola is F(1,0), and the equation of the directrix is x=-1. Using the definition of a parabola, we can find the value of x_{1}+x_{2}, which then allows us to calculate the distance from M to the directrix as dfrac {x_{1}+x_{2}}{2}+1. This problem primarily tests your understanding of the definition, standard equation, and basic properties of parabolas and is considered to be of medium difficulty.