answer:To find the percentage increase in state tax, we first need to determine the difference in cost between buying the bike this month and next month. The cost next month is 82,500, and the cost this month is 75,000. The difference in cost is 82,500 - 75,000 = 7,500. This 7,500 increase is due to the state tax increase alone. To find the percentage increase, we divide the increase by the original cost (this month's cost) and then multiply by 100 to get the percentage. Percentage increase = (7,500 / 75,000) * 100 Percentage increase = 0.1 * 100 Percentage increase = 10% So, the state tax will increase by boxed{10%} next month.

answer:1. **Position of the hands**: At 3:15 p.m., the minute hand points at the 3 (or 15 minutes), and the hour hand is a quarter of the way between 3 and 4. 2. **Calculate the hour hand's angular position**: The hour hand moves at a rate of 30^circ per hour (since 360^circ / 12 = 30^circ). At 3:00 it would be at 3 times 30^circ = 90^circ from the top (12:00). In 15 minutes, it moves an additional frac{15}{60} times 30^circ = 7.5^circ. Thus, at 3:15, the hour hand is at 90^circ + 7.5^circ = 97.5^circ. 3. **Calculate the minute hand's angular position**: The minute hand moves at a rate of 360^circ per hour, or 6^circ per minute (since 360^circ / 60 = 6^circ). At 15 minutes, it points at 90 degrees (as it points directly at the 3). 4. **Calculate the angle between the hands**: The difference between the hour and minute hand positions at 3:15 is |97.5^circ - 90^circ| = 7.5^circ. Conclusion: The smaller angle formed by the minute hand and the hour hand at 3:15 p.m. is boxed{7.5^circ}.

answer:Split the sum and simplify by grouping similar terms: [ sum^{50}_{i=1} sum^{50}_{j=1} (2i + 2j + 3) = sum^{50}_{i=1} sum^{50}_{j=1} 2i + sum^{50}_{i=1} sum^{50}_{j=1} 2j + sum^{50}_{i=1} sum^{50}_{j=1} 3 ] As i and j are constant within their respective sums over the other variable, and there are 50 terms in each summation: [ = 2left(50 sum^{50}_{i=1} i + 50 sum^{50}_{j=1} jright) + 50^2 cdot 3 ] The sum of the first n natural numbers sum^{n}_{k=1} k = frac{n(n+1)}{2}. Substituting n = 50: [ sum^{50}_{k=1} k = frac{50 times 51}{2} = 1275 ] Substitute back into the equation: [ = 2(50 cdot 1275 + 50 cdot 1275) + 50^2 cdot 3 = 2(127500) + 7500 = 255000 + 7500 = 262500 ] Thus, the sum is: [ 262500 ] The final answer is boxed{C) 262500}

answer:: 1. **Identify Key Points and Parameters**: Let's denote the points ( A, B, C, D ) of the tetrahedron ( ABCD ) with their perpendicular drops as ( A_1, B_1, C_1, D_1 ) intersecting at point ( H ). There are additional points ( A_2, B_2, C_2 ) on the altitudes such that ( AA_2: A_2A_1 = BB_2: B_2B_1 = CC_2: C_2C_1 = 2:1 ). 2. **Define Median Intersection**: Let point ( M ) be the centroid of ( triangle ABC ), defined as the intersection of the medians from ( A, B, C ). 3. **Establish Proportions and Similarity**: Considering ( AA_3 ) and ( CC_3 ) are medians, and using the given ratios: [ MA: A_3A = AA_2: A_1A = 2:3 ] Also, from the angles ( angle MAA_2 ) and ( angle A_3AA_1 ): [ triangle MAA_2 sim triangle A_3AA_1 implies frac{MA_2}{A_3A_1} = frac{2}{3} ] 4. **Parallel and Perpendicular Conditions**: Hence ( MA_2 parallel A_3A_1 ). Given ( AA_2 perp text{plane } BCD implies AA_1 perp A_3A_1 ), thus ( MA_2 perp A_1A ). This implies ( angle MA_2H = 90^circ ). 5. **Similar Conditions for Other Altitudes**: Analogously, using similar arguments: [ angle MB_2H = 90^circ quad text{and} quad angle MC_2H = 90^circ ] 6. **Perpendicularity of ( DD_1 )**: Since ( D_1 ) is the foot of the altitude from ( D ) in the tetrahedron, ( DD_1 perp MD_1 ), which implies: [ angle MD_1H = 90^circ ] 7. **Conclusion**: Since all the angles ( angle MA_2H, angle MB_2H, angle MC_2H, angle MD_1H ) are right angles, the points ( M, A_2, B_2, C_2, D_1 ), and ( H ) lie on a sphere with diameter ( MH ). Therefore, ( H, A_2, B_2, C_2, D_1 ) are concyclic, lying on the same sphere. [ boxed{} ]