answer:1. **List Price Setup**: Let the list price of the goods be (L = 100) units for simplicity. 2. **Calculate the Purchase Price**: With a 30% discount, the purchase price is: [ 100 - 30% times 100 = 100 - 30 = 70 text{ units} ] 3. **Let the Marked Price be (x) units**. 4. **Determine the Selling Price**: The selling price after a 20% discount on the marked price is: [ x - 20% times x = 0.8x ] 5. **Profit Requirement**: The desired profit is now 30% of the selling price. Setting up the profit equation: [ 0.8x - 70 = 0.3 times 0.8x ] 6. **Solving for (x)**: [ 0.8x - 70 = 0.24x ] [ 0.8x - 0.24x = 70 ] [ 0.56x = 70 ] [ x = frac{70}{0.56} approx 125 ] 7. **Conclusion**: Therefore, the marked price (x) is about 125 units. As a percentage of the list price (L), which is 100 units: [ frac{125}{100} times 100% = 125% ] Thus, the merchant must mark the goods at 125% of the list price. The final answer is boxed{125%} (Option A).

answer:To prove this, let g(x)=f(x)-x. Since g(0)=frac{1}{4} and gleft(frac{1}{2}right)=fleft(frac{1}{2}right)-frac{1}{2}=-frac{1}{8}, it follows that g(0) cdot gleft(frac{1}{2}right) < 0. Furthermore, the function g(x) is a continuous curve on the interval left[0,frac{1}{2}right], therefore, there exists x_{0} in (0, frac{1}{2}), such that g(x_{0})=0, which means f(x_{0})=x_{0}. Thus, we have boxed{text{There exists } x_{0} in left( 0, frac{1}{2} right) text{ such that } f(x_{0})=x_{0}.}

answer:To find the minimum amount of grass seed the customer needs to buy, we need to determine the combination of bags that will cost 98.75 while not exceeding 80 pounds. First, let's calculate the cost per pound for each bag size: - 5-pound bag: 13.82 / 5 pounds = 2.764 per pound - 10-pound bag: 20.43 / 10 pounds = 2.043 per pound - 25-pound bag: 32.25 / 25 pounds = 1.29 per pound Since the 25-pound bag offers the lowest cost per pound, the customer should buy as many of these as possible without exceeding 80 pounds. Let's see how many 25-pound bags can be bought for 98.75: Number of 25-pound bags = 98.75 / 32.25 per bag = approximately 3.06 bags Since the customer cannot buy a fraction of a bag, they can buy at most 3 bags of the 25-pound size, which would cost: 3 bags * 32.25 per bag = 96.75 Now, we have 98.75 - 96.75 = 2.00 left to spend, and we need to buy more grass seed without exceeding 80 pounds. The customer has already bought 75 pounds (3 bags * 25 pounds), so they can buy up to 5 more pounds. Since the 5-pound bag is the smallest size and the customer can only buy whole bags, they cannot use the remaining 2.00 to buy another bag because the 5-pound bag costs 13.82, which is more than 2.00. Therefore, the minimum amount of grass seed the customer needs to buy is the amount they can get with 96.75, which is boxed{75} pounds from the three 25-pound bags.

answer:1. **Identify the Range for the Fourth Rod**: To form a quadrilateral, the sum of the lengths of any three sides must be greater than the length of the fourth side. 2. **Maximum Possible Length for the Fourth Rod**: [ 5 + 12 + 25 = 42 ] The fourth rod must be less than 42 cm to satisfy the triangle inequality with the total length of the other three rods. 3. **Minimum Possible Length for the Fourth Rod**: [ 25 - (5 + 12) = 25 - 17 = 8 ] The fourth rod must be greater than 8 cm to ensure the sum of the lengths of the three smaller rods (including the fourth rod) is greater than the length of the longest rod (25 cm). 4. **Determine the Valid Lengths for the Fourth Rod**: The fourth rod's length must be between 9 cm to 41 cm inclusive. 5. **Count the Number of Valid Rods**: The integers from 9 to 41 inclusive are: [ 9, 10, 11, ldots, 41 ] The total number of integers in this range is 41 - 9 + 1 = 33. 6. **Exclude the Rods Already Used**: The rods of lengths 5 cm, 12 cm, and 25 cm are already used and cannot be chosen again. 7. **Conclusion**: There are 33 rods that Joy can choose as the fourth rod to form a quadrilateral with positive area. Thus, the answer is 33. The final answer is boxed{D) 33}.