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question:Matt, Alex, and their friends John and Kate decide to wash their cars. Matt spends 80 minutes washing the outside of his car and 1/4 that time cleaning the inside. Alex spends half as much time as Matt cleaning the outside of his car, but twice as much time as Matt cleaning the inside. John takes 50% more time than Matt to clean the outside, but 75% of Matt's time for the inside. Kate cleans the outside of her car in the same time as Alex but takes 20 minutes more than the average time of the other three friends to clean the inside. Additionally, they all take a 10minute break before cleaning the inside of their cars. Calculate the total time Matt, Alex, John, and Kate spend cleaning their cars, including the breaks.
answer:Let's calculate the time each person spends on cleaning their cars, including the breaks. **Matt:** - Outside: 80 minutes - Break: 10 minutes - Inside: 1/4 of 80 minutes = 80 / 4 = 20 minutes - Total for Matt: 80 + 10 + 20 = 110 minutes **Alex:** - Outside: 1/2 of Matt's time = 1/2 * 80 = 40 minutes - Break: 10 minutes - Inside: 2 times Matt's time = 2 * 20 = 40 minutes - Total for Alex: 40 + 10 + 40 = 90 minutes **John:** - Outside: 50% more than Matt = 1.5 * 80 = 120 minutes - Break: 10 minutes - Inside: 75% of Matt's time = 0.75 * 20 = 15 minutes - Total for John: 120 + 10 + 15 = 145 minutes **Kate:** - Outside: Same as Alex = 40 minutes - Break: 10 minutes - Inside: We need to calculate the average time of the other three friends first. Average time for the inside (excluding Kate): - Matt's inside time: 20 minutes - Alex's inside time: 40 minutes - John's inside time: 15 minutes - Average = (20 + 40 + 15) / 3 = 75 / 3 = 25 minutes Kate's inside time: Average + 20 minutes = 25 + 20 = 45 minutes Total for Kate: 40 + 10 + 45 = 95 minutes **Total time for all four:** - Matt: 110 minutes - Alex: 90 minutes - John: 145 minutes - Kate: 95 minutes Total = 110 + 90 + 145 + 95 = 440 minutes So, Matt, Alex, John, and Kate spend a total of boxed{440} minutes cleaning their cars, including the breaks.
question:Given the sequence a_n=n^{2}+9n+10, find the maximum value of n for which the sum of the first n terms is maximized. A: 4 B: 5 C: 9 or 10 D: 4 or 5
answer:We have a_n=-n^{2}+9n+10=-(n-10)(n+1). Since the sum of the first n terms, denoted by S_n, has a maximum value, we have S_n geq S_{n+1}. This implies that a_{n+1} leq 0, which gives us -[(n+1)-10][(n+1)+1] leq 0. Solving this inequality, we get n geq 9. We also have a_8=18, a_9=10, a_{10}=0, and a_{11}=-12. Therefore, S_9=S_{10} is the maximum sum, and this occurs when n=9 or 10. Hence, the answer is boxed{text{C}}. From the problem, we can deduce that S_n geq S_{n+1}. By solving this inequality and considering the signs of the terms, we can make a judgment. This problem tests the ability to find the general term formula of a sequence and the method for finding the maximum value of the sum of the first n terms of a sequence. When solving the problem, it is necessary to reasonably apply the formulas and methods.
question:Given a circle mathcal{C} with center O and a point A outside the circle. From point A, two rays are drawn: one intersects the circle at points B and C (in that order) and the other intersects the circle at points D and E (also in that order). Show that widehat{C A E}=frac{widehat{C O E}widehat{B O D}}{2}
answer:1. **Identify Key Elements and Theorems**: - Let ( mathcal{C} ) be a circle centered at ( O ). - Point ( A ) is outside the circle. - Points ( B ) and ( C ) are points where one ray from ( A ) intersects the circle. - Points ( D ) and ( E ) are points where another ray from ( A ) intersects the circle. 2. **Apply Inscribed Angle Theorem**: - According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of the central angle that subtends the same arc. - Consequently, [ frac{widehat{C O E}}{2} = widehat{C B E} quad text{(using the arc (CE))}. ] 3. **Evaluate Inscribed Angles**: - Similarly, [ frac{widehat{B O D}}{2} = widehat{B E D} quad text{(using the arc (BD))}. ] 4. **Substitute and Simplify**: - We are given the expression (frac{widehat{C O E} - widehat{B O D}}{2}), and substituting the above equalities gives: [ frac{widehat{C O E} - widehat{B O D}}{2} = widehat{C B E} - widehat{B E D}. ] 5. **Angle Relationships**: - Notice that (widehat{C B E} + widehat{B E A} = pi) (straight line theorem at point ( E )). - Thus, [ widehat{C B E} = pi - widehat{B E A}. ] - Therefore: [ widehat{C B E} - widehat{B E D} = (pi - widehat{B E A}) - widehat{B E A} = pi - 2 widehat{B E A}. ] 6. **Final Geometric Simplification**: - Recall the external angle theorem: (pi = widehat{C A E} + widehat{B E A} + widehat{A B E}), which can be rearranged as: [ widehat{C A E} = pi - widehat{B E A} - widehat{A B E}. ] 7. **Combining Results**: - Since (widehat{A B E}) is the additional angle inside the triangle ( triangle ABE ), we adjust: [ boxed{frac{widehat{C O E} - widehat{B O D}}{2} = widehat{C A E}}. ] Thus, we have shown that (boxed{frac{widehat{C O E} - widehat{B O D}}{2} = widehat{C A E}}).
question:f the radius of a circle is 1, then the arc length corresponding to a 60^{circ} central angle is ( ) A: frac{π}{2} B: pi C: frac{π}{6} D: frac{π}{3}
answer:To solve for the arc length l corresponding to a 60^{circ} central angle in a circle with radius 1, we use the formula for arc length in terms of the central angle in degrees, which is l = frac{npi r}{180}. Substituting n = 60^{circ} and r = 1, we get: [ l = frac{60pi cdot 1}{180} = frac{60pi}{180} = frac{pi}{3} ] Therefore, the arc length corresponding to a 60^{circ} central angle in a circle with radius 1 is boxed{text{D: } frac{pi}{3}}.