answer:To find the smallest positive integer g such that 3150 multiplied by g is the square of an integer, we need to prime factorize 3150 and then determine what factors are needed to make it a perfect square. Let's start by prime factorizing 3150: 3150 = 2 × 3 × 3 × 5 × 5 × 7 3150 = 2 × 3^2 × 5^2 × 7 For a number to be a perfect square, all the prime factors must be raised to an even power. In the prime factorization of 3150, the primes 2 and 7 are not squared. To make 3150 a perfect square, we need to multiply it by a number that will square these primes. So, we need to multiply 3150 by 2 × 7 to make it a perfect square: g = 2 × 7 = 14 Now, let's check if 3150 × 14 is indeed a perfect square: 3150 × 14 = 2 × 3^2 × 5^2 × 7 × 2 × 7 3150 × 14 = 2^2 × 3^2 × 5^2 × 7^2 Now, every prime factor is raised to an even power, which means 3150 × 14 is a perfect square. Therefore, the smallest positive integer g is boxed{14} .

answer:First, apply the Pythagorean Theorem to triangle ABC: [AB = sqrt{AC^2 + BC^2} = sqrt{9^2 + 12^2} = sqrt{81 + 144} = sqrt{225} = 15.] Since triangle DBEsimtriangle ABC (by AA similarity, as angle DBE = angle BAC = 90^circ and angle BED = angle ABC), the ratio of corresponding sides must be equal. Thus: [frac{BD}{BA} = frac{DE}{AC}.] Substitute the known values: [BD = frac{DE}{AC} times BA = frac{6}{9} times 15 = frac{2}{3} times 15 = 10.] Conclusively: [boxed{BD = 10}.]

answer:To solve this problem, we start by defining variables for the quantities we're dealing with. Let x represent the number of type A classified trash cans purchased. Since the total number of trash cans to be purchased is 8, the number of type B classified trash cans purchased will be 8-x. Given the cost of each type A trash can is 150 yuan and each type B trash can is 225 yuan, we can set up an inequality to represent the total cost constraint, which should not exceed 1500 yuan. The inequality is as follows: [150x + 225(8-x) leq 1500] Expanding and simplifying this inequality gives us: [150x + 1800 - 225x leq 1500] [1800 - 75x leq 1500] [300 geq 75x] [4 geq x] However, since x represents the number of type A trash cans, and we know we need to purchase a total of 8 trash cans, x must also satisfy the condition that 8-x (the number of type B trash cans) is a non-negative integer. This means x can take on values where it is less than or equal to 8 but also greater than or equal to 4 to ensure we do not exceed the budget. Therefore, x can be 4, 5, 6, 7, or 8. Counting these options, we find there are a total of boxed{5} different ways to make the purchase, which corresponds to choice C.

answer:To calculate the overall loss percentage for the store, we first need to determine the total cost price of all the items, the total selling price of all the items, and then the total loss. After that, we can calculate the loss percentage. 1. Total Cost Price (TCP) of all items: TCP = Cost of radio + Cost of television + Cost of refrigerator TCP = Rs. 1500 + Rs. 8000 + Rs. 25000 TCP = Rs. 34500 2. Total Selling Price (TSP) of all items: TSP = Selling price of radio + Selling price of television + Selling price of refrigerator TSP = Rs. 1110 + Rs. 7500 + Rs. 23000 TSP = Rs. 31610 3. Total Loss (TL): TL = TCP - TSP TL = Rs. 34500 - Rs. 31610 TL = Rs. 2890 4. Loss Percentage (LP): LP = (Total Loss / Total Cost Price) * 100 LP = (Rs. 2890 / Rs. 34500) * 100 LP = 0.08376811594 * 100 LP = 8.376811594% Therefore, the overall loss percentage for the store is approximately boxed{8.38%} .