answer:To find the next palindromic year after 2021, we sequentially check the years following 2021. - **Check year 2022**: Not a palindrome. - **Check year 2023**: Not a palindrome. - **Check year 2024**: Not a palindrome. - **Check year 2025**: Not a palindrome. - **Check year 2026**: Not a palindrome. - **Check year 2027**: Not a palindrome. - **Check year 2028**: Not a palindrome. - **Check year 2029**: Not a palindrome. - **Check year 2030**: Not a palindrome. - **Check year 2031**: Not a palindrome. - **Check year 2121**: Reads the same forward (2121) and backward (2121), so it is a palindrome. Next, we calculate the product of the digits in the year 2121. - The digits of 2121 are 2, 1, 2, and 1. - The product of these digits is calculated as: [ 2 times 1 times 2 times 1 = 4 ] # Conclusion: The product of the digits of the next palindromic year after 2021 is 4. The correct answer is boxed{text{(B)} 4}.

answer:If each necklace cost 40,000, then the total cost for the three necklaces would be: 3 necklaces * 40,000 per necklace = 120,000 Since the earrings were three times as expensive as one necklace, their cost would be: 3 * 40,000 = 120,000 Now, to find the total cost of the necklaces and earrings, we add the cost of the necklaces to the cost of the earrings: Total cost = Cost of necklaces + Cost of earrings Total cost = 120,000 + 120,000 Total cost = 240,000 Therefore, the total cost of the necklaces and earrings was boxed{240,000} .

answer:1. **Understanding the Pyramid Structure**: - We are given a pyramid (M-ABCD) with a square base (ABCD). - It is given that (MA = MD) and (MA perp AB). - The area of triangle (AMD) is 1. 2. **Finding the Lengths and Heights**: - Let us denote (AD = a). - (MA) can be considered as the height of the right triangle (AMD). - Since (MA = MD) and they are both heights from (M) to the sides of (ABCD), (triangle MAD) is isosceles and has a right angle. 3. **Area of Triangle (AMD)**: - The area of (triangle AMD) is given by: [ text{Area} = frac{1}{2} times AD times MA = 1 ] - Substituting (AD = a) and let (MA = x): [ frac{1}{2} times a times x = 1 implies a times x = 2 implies x = frac{2}{a} ] 4. **Length of (ME)**: - The length (ME) can be determined by recognizing that (E) is the midpoint of (AD). Therefore, according to the relationships in the right triangles: [ ME = frac{2}{a} ] 5. **Finding the Radius (r)**: - Let (r) be the radius of the largest sphere that can fit inside the pyramid (M-ABCD). - Using the formula for the inradius of a right triangle: [ r = frac{ME + EF - MF}{2} ] Since (E) is the midpoint, (EF = frac{a}{2}), and by the Pythagorean theorem: [ MF = sqrt{ (ME)^2 + (EF)^2 } = sqrt{ left(frac{2}{a}right)^2 + left(frac{a}{2}right)^2 } = sqrt{ frac{4}{a^2} + frac{a^2}{4} } ] - Simplify the expression inside the square root: [ MF = sqrt{ frac{4}{a^2} + frac{a^2}{4}} = sqrt{ frac{4}{a^2} + frac{a^2 * 4}{16a^2} } = sqrt{ frac{4 + a^2}{4}} ] - Therefore, [ r = frac{a + frac{2}{a} - MF}{2} = frac{a + frac{2}{a} - sqrt{ frac{4 + a^2}{4} }}{2} ] 6. **Maximize the Radius**: - The expression for (r) needs to be simplified further: [ r = frac{a + frac{2}{a} - frac{sqrt{ 4 + a^2}}{2}}{2} ] - Given the geometric properties, the radius of the sphere will achieve the maximum value when (a = sqrt{2}). Plugging (a = sqrt{2}) into the expression, simplifies it to: [ r = frac{a + frac{2}{a} - sqrt{ 2}}{2} = sqrt{2}-1 ] In conclusion, the radius of the largest sphere that can fit inside the pyramid is: [ boxed{sqrt{2} - 1} ]

answer:1. **Understanding the problem**: Originally, Paula had enough paint for 50 rooms, but after losing five cans, she could only paint 40 rooms. We need to find out the total number of cans she had initially. 2. **Relating cans of paint to rooms**: The loss of five cans resulted in a decrease of 10 rooms that could be painted. Thus, each can of paint is sufficient to paint frac{10}{5} = 2 rooms. 3. **Calculating the number of cans for 50 rooms**: Since each can paints 2 rooms, to paint 50 rooms she would have needed: [ text{Number of cans} = frac{50 text{ rooms}}{2 text{ rooms per can}} = 25 ] 4. **Conclusion**: Paula originally had 25 cans of paint. Thus, the final answer is 25. The final answer is boxed{C}