answer:First, we simplify the given complex fraction by multiplying both the numerator and denominator by the conjugate of the denominator: z= frac {3+4i}{1-2i} = frac {(3+4i)(1+2i)}{(1-2i)(1+2i)} = frac {-5+10i}{5} = -1+2i. Then, we find the conjugate of z, denoted as overline{z}, which is simply -1 - 2i. Finally, we compute the modulus of overline{z}, denoted as | overline{z}|, using the formula for the modulus of a complex number: | overline{z}| = sqrt {(-1)^{2}+(-2)^{2}} = sqrt {5}. Therefore, the final answer is: boxed{sqrt {5}}.

answer:1. **Calculate the volume of the cube**: With a side length of 5, the volume of the cube (V_{text{cube}}) is: [ V_{text{cube}} = 5^3 = 125. ] 2. **Calculate the volume of a ball of clay**: Each ball of clay has a radius of 1. The volume of a sphere (V_{text{ball}}) is given by the formula: [ V_{text{ball}} = frac{4}{3} pi (1)^3 = frac{4}{3} pi. ] 3. **Determine the maximum number of spherical balls that can fit in the cube**: The number of balls that can fit is determined by dividing the cube's volume by the volume of one ball, rounded down: [ text{Number of balls} = leftlfloor frac{V_{text{cube}}}{V_{text{ball}}} rightrfloor = leftlfloor frac{125}{frac{4}{3} pi} rightrfloor = leftlfloor frac{375}{4 pi} rightrfloor. ] 4. **Estimation using pi approx 3.14**: [ 4pi approx 12.56, ] and thus: [ leftlfloor frac{375}{12.56} rightrfloor approx leftlfloor 29.87 rightrfloor = 29. ] 5. **Conclusion**: The maximum number of spherical balls of clay that can completely fit inside the cube is 29. The final answer is The correct final answer, given the choices is boxed{textbf{(C)} text{29}}.

answer:To find the difference between the largest and smallest geometric numbers with 3 distinct digits forming a geometric sequence, let's follow the solution closely: 1. **Identifying the Largest Geometric Number:** - The largest digit that can start a 3-digit number is 9. - For the sequence to be geometric with distinct digits and the third term also being a whole number, the common ratio should be a fraction of the form frac{k}{3}, where k is an integer. - When k=1, the sequence is 9, 3, 1, forming the number 931. - When k=2, the sequence is 9, 6, 4, forming the number 964. - When k=3, the sequence would be 9, 9, 9, which does not meet the requirement of distinct digits. - Therefore, the largest geometric number is 964. 2. **Identifying the Smallest Geometric Number:** - By applying similar logic in reverse, the smallest geometric number with distinct digits forming a geometric sequence is found to be 124. 3. **Calculating the Difference:** - The difference between the largest and smallest geometric numbers is 964 - 124. - This simplifies to 840. Therefore, the difference between the largest and smallest geometric numbers is boxed{840}.

answer:To calculate the total cost of Hilary's meal at the Delicious Delhi restaurant, including the tip, we follow these steps: 1. First, calculate the cost of the samosas. Hilary bought three samosas at 2 each, so the cost for the samosas is: [ 3 times 2 = 6 ] 2. Next, calculate the cost of the pakoras. Hilary bought four orders of pakoras at 3 each, so the cost for the pakoras is: [ 4 times 3 = 12 ] 3. Add the cost of a mango lassi, which is 2, to the cost of the samosas and pakoras to find the total cost of the food before the tip: [ 6 + 12 + 2 = 20 ] 4. Calculate the tip by taking 25% of the total food cost. The tip is: [ 20 times 0.25 = 5 ] 5. Finally, add the tip to the total cost of the food to find the total cost of the meal, including the tip: [ 20 + 5 = 25 ] Therefore, the total cost of Hilary's meal, including the tip, is boxed{25} dollars.