| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To find the total number of supporters the candidate will have after 60 days, we need to integrate the rate function over the time interval from 0 to 60 days. The rate function is given by: r(t) = 2000t + 1000 We want to find the total number of supporters (votes) gained from t = 0 to t = 60. This can be found by integrating the rate function with respect to time (t) over the interval [0, 60]. The integral of the rate function r(t) = 2000t + 1000 from 0 to 60 is: ∫(2000t + 1000)dt from 0 to 60 = [1000t^2 + 1000t] evaluated from 0 to 60 = (1000 * 60^2 + 1000 * 60) - (1000 * 0^2 + 1000 * 0) = (1000 * 3600 + 1000 * 60) = 3600000 + 60000 = 3660000 Therefore, the candidate will have gained a total of 3,660,000 supporters after 60 days, assuming she had no supporters at t = 0. |
3660000 |
| deepseekr1 | To determine the total number of supporters the mayoral candidate will have after 60 days, we integrate the rate function \( 2000t + 1000 \) votes per day from \( t = 0 \) to \( t = 60 \): \[ \text{Total Votes} = \int_{0}^{60} (2000t + 1000) \, dt \] **Step 1: Integrate the rate function** \[ \int (2000t + 1000) \, dt = 1000t^2 + 1000t + C \] **Step 2: Evaluate the definite integral from 0 to 60** \[ \left[ 1000t^2 + 1000t \right]_{0}^{60} = \left( 1000(60)^2 + 1000(60) \right) - \left( 1000(0)^2 + 1000(0) \right) \] **Step 3: Calculate the result** \[ = 1000(3600) + 1000(60) = 3,600,000 + 60,000 = 3,660,000 \] **Final Answer:** The candidate will have \(\boxed{3,660,000}\) supporters after 60 days. |
3660000 |
| gpto1 | To find out how many supporters the candidate will have after 60 days, we'll need to calculate the total number of votes she gains over that period. The rate at which she is gaining votes is given by the function: \[ R(t) = 2000t + 1000 \] where: - \( R(t) \) is the rate of gaining votes per day. - \( t \) is the number of days since she announced her candidacy. Since we know the rate of change, we can find the total number of supporters by integrating the rate function over the interval from \( t = 0 \) to \( t = 60 \) days. ### Step 1: Set Up the Integral The total number of supporters, \( S \), after 60 days is: \[ S = \int_{0}^{60} R(t) \, dt = \int_{0}^{60} (2000t + 1000) \, dt \] ### Step 2: Integrate the Function We'll integrate each term of the function separately. #### Integrate \( 2000t \): \[ \int_{0}^{60} 2000t \, dt = 2000 \int_{0}^{60} t \, dt = 2000 \left[ \frac{1}{2}t^2 \right]_{0}^{60} \] #### Integrate \( 1000 \): \[ \int_{0}^{60} 1000 \, dt = 1000 \int_{0}^{60} dt = 1000 \left[ t \right]_{0}^{60} \] ### Step 3: Evaluate the Integrals #### Evaluate the first integral: \[ 2000 \left[ \frac{1}{2}t^2 \right]_{0}^{60} = 2000 \left( \frac{1}{2}(60)^2 - \frac{1}{2}(0)^2 \right) \] \[ = 2000 \left( \frac{1}{2}(3600) - 0 \right) = 2000 \left( 1800 \right) = 3,600,000 \] #### Evaluate the second integral: \[ 1000 \left[ t \right]_{0}^{60} = 1000 (60 - 0) = 1000 \times 60 = 60,000 \] ### Step 4: Add the Results \[ S = \text{First Integral} + \text{Second Integral} = 3,600,000 + 60,000 = 3,660,000 \] ### Conclusion After 60 days, the candidate will have **3,660,000 supporters**. |
3660000 |
| GPT-4o | To determine the number of supporters the mayoral candidate will have after 60 days, we need to integrate the given rate of gaining votes with respect to time. Given the rate of gaining votes per day: \( \frac{dV}{dt} = 2000t + 1000 \) To find the total number of votes \( V(t) \) after \( t \) days, we integrate the rate function: \[ V(t) = \int (2000t + 1000) \, dt \] Let's compute the integral: \[ V(t) = \int 2000t \, dt + \int 1000 \, dt \] For the first term: \[ \int 2000t \, dt = 2000 \int t \, dt = 2000 \left( \frac{t^2}{2} \right) = 1000t^2 \] For the second term: \[ \int 1000 \, dt = 1000t \] Combining these results, we get: \[ V(t) = 1000t^2 + 1000t + C \] Given that the candidate had no supporters at \( t = 0 \), we use the initial condition \( V(0) = 0 \) to find the constant \( C \): \[ V(0) = 1000(0)^2 + 1000(0) + C = 0 \] \[ C = 0 \] Thus, the equation for the number of supporters becomes: \[ V(t) = 1000t^2 + 1000t \] Now, we need to find the number of supporters after 60 days \( (t = 60) \): \[ V(60) = 1000(60)^2 + 1000(60) \] \[ V(60) = 1000(3600) + 1000(60) \] \[ V(60) = 3600000 + 60000 \] \[ V(60) = 3660000 \] Therefore, the candidate will have \( 3,660,000 \) supporters after 60 days. |
3660000 |
| llama3.1-70B-Instruct | To find the total number of supporters the candidate will have after 60 days, we need to integrate the rate of gain of votes over the 60-day period. Let's denote the number of supporters as S(t) and the rate of gain of votes as R(t) = 2000t + 1000. We know that the rate of change of the number of supporters is equal to the rate of gain of votes: dS/dt = R(t) = 2000t + 1000 To find the total number of supporters after 60 days, we need to integrate the rate of gain of votes from t = 0 to t = 60: S(60) = ∫[0,60] R(t) dt = ∫[0,60] (2000t + 1000) dt Evaluating the integral, we get: S(60) = [1000t^2 + 1000t] from 0 to 60 = (1000(60)^2 + 1000(60)) - (1000(0)^2 + 1000(0)) = 3600000 + 60000 = 3660000 Therefore, the candidate will have approximately 3,660,000 supporters after 60 days, assuming she had no supporters at t = 0. |
3660000 |
| llama3.1-405B-Instruct | To find the number of supporters the candidate will have after 60 days, we need to integrate the rate of change of supporters with respect to time. The rate of change of supporters is given by: dS/dt = 2000t + 1000 where S is the number of supporters and t is the number of days. To find the number of supporters after 60 days, we need to integrate this rate of change from t = 0 to t = 60: S(60) = ∫[0,60] (2000t + 1000) dt = [2000/2]t^2 + 1000t | [0,60] = 1000(60)^2 + 1000(60) = 3600000 + 60000 = 3660000 Therefore, the candidate will have approximately 3,660,000 supporters after 60 days, assuming that she had no supporters at t = 0. |
3660000 |
| DeepSeek-Coder-V2 | To solve the problem, we need to determine the total number of supporters the candidate will have after 60 days, given the rate of gaining votes. 1. **Understand the rate of gaining votes**: The rate at which the candidate is gaining votes is given by the function 2000t + 1000 votes per day, where t is the number of days since she announced her candidacy. 2. **Set up the integral**: To find the total number of supporters after 60 days, we need to integrate the rate function over the interval from t = 0 to t = 60. The integral to solve is: \[ \int_{0}^{60} (2000t + 1000) \, dt \] 3. **Evaluate the integral**: \[ \int (2000t + 1000) \, dt = 2000 \int t \, dt + 1000 \int 1 \, dt \] Solving each part separately: \[ 2000 \int t \, dt = 2000 \left( \frac{t^2}{2} \right) = 1000t^2 \] \[ 1000 \int 1 \, dt = 1000t \] Combining the results: \[ \int (2000t + 1000) \, dt = 1000t^2 + 1000t \] 4. **Apply the limits of integration**: \[ \left[ 1000t^2 + 1000t \right]_{0}^{60} \] Substitute t = 60 and t = 0: \[ \left( 1000(60)^2 + 1000(60) \right) - \left( 1000(0)^2 + 1000(0) \right) \] \[ = 1000(3600) + 1000(60) - 0 \] \[ = 3600000 + 60000 \] \[ = 3660000 \] 5. **Conclusion**: After 60 days, the candidate will have 3,660,000 supporters. Thus, the solution to the problem is: \[ \boxed{3660000} \] |
3660000 |
| Qwen2-72B-Instruct | To find the total number of supporters the candidate will have after 60 days, we need to integrate the rate function over the time interval from 0 to 60 days. The rate function is given by: r(t) = 2000t + 1000 We want to find the total number of supporters (votes) gained from t = 0 to t = 60. This can be found by integrating the rate function with respect to time (t) over the interval [0, 60]. The integral of the rate function r(t) = 2000t + 1000 from 0 to 60 is: ∫(2000t + 1000)dt from 0 to 60 = [1000t^2 + 1000t] evaluated from 0 to 60 = (1000 * 60^2 + 1000 * 60) - (1000 * 0^2 + 1000 * 0) = (1000 * 3600 + 1000 * 60) = 3600000 + 60000 = 3660000 Therefore, the candidate will have gained a total of 3,660,000 supporters after 60 days, assuming she had no supporters at t = 0. |
3660000 |
| baichuan4 | To solve this problem, we need to calculate the number of supporters the candidate will have after 60 days by plugging the value of t (60 days) into the given equation for the rate of gaining votes. The equation provided is: Votes gained per day = 2000t + 1000 Where t is the number of days since the candidacy announcement. Let's plug in the value of t = 60 days: Votes gained in 60 days = 2000(60) + 1000 Calculating the result: Votes gained in 60 days = 120000 + 1000 Votes gained in 60 days = 121000 So, after 60 days, the candidate will have gained 121,000 supporters, assuming she had no supporters at t = 0. |
3660000 |
| doubao | To find the total number of supporters after 60 days, we need to integrate the rate function from 0 to 60. The integral of 2000t + 1000 with respect to t is 1000t^2 + 1000t. Evaluating this from 0 to 60: [1000(60)^2 + 1000(60)] - [1000(0)^2 + 1000(0)] = [1000(3600) + 60000] - 0 = 3600000 + 60000 = 3660000. So the candidate will have 3,660,000 supporters after 60 days. |
3660000 |
| minimax | To find out how many supporters the mayoral candidate will have after 60 days, we need to integrate the rate of gaining votes over the time period from t = 0 to t = 60. The rate of gaining votes is given by the function: \[ V(t) = 2000t + 1000 \] To find the total number of votes after 60 days, we integrate this function from 0 to 60: \[ \text{Total Votes} = \int_{0}^{60} (2000t + 1000) \, dt \] \[ \text{Total Votes} = \left[ \frac{2000}{2}t^2 + 1000t \right]_{0}^{60} \] \[ \text{Total Votes} = \left[ 1000t^2 + 1000t \right]_{0}^{60} \] Now we evaluate the expression at the upper limit (t = 60) and subtract the value at the lower limit (t = 0): \[ \text{Total Votes} = \left( 1000 \cdot 60^2 + 1000 \cdot 60 \right) - \left( 1000 \cdot 0^2 + 1000 \cdot 0 \right) \] \[ \text{Total Votes} = \left( 1000 \cdot 3600 + 1000 \cdot 60 \right) - (0) \] \[ \text{Total Votes} = 3600000 + 60000 \] \[ \text{Total Votes} = 3660000 \] So, after 60 days, the mayoral candidate will have 3,660,000 supporters. |
3660000 |
| kimi | To find out how many supporters the candidate will have after 60 days, we need to calculate the total number of votes she gains over the 60 days using the given rate. The rate of votes gained per day is given by the function: \[ V(t) = 2000t + 1000 \] Where: - \( V(t) \) is the number of votes at day \( t \). - \( t \) is the number of days since the candidate announced her candidacy. We want to find the total number of votes \( V(60) \) after 60 days. First, we plug in \( t = 60 \) into the function: \[ V(60) = 2000 \times 60 + 1000 \] Now, we calculate the value: \[ V(60) = 120000 + 1000 \] \[ V(60) = 121000 \] So, after 60 days, the candidate will have 121,000 supporters. |
3660000 |
| wenxinyiyan | To find the total number of supporters the candidate will have after 60 days, we need to calculate the area under the curve represented by the function 2000t + 1000 from t = 0 to t = 60. This area represents the cumulative number of votes gained over that period. The function representing the daily vote gain is a linear function of t, so we can use the formula for the area of a trapezoid to approximate the area under the curve. However, since this is a linear function, we can also directly calculate the integral to find the exact area. The integral of 2000t + 1000 with respect to t is: ∫(2000t + 1000) dt = 1000t^2 + 1000t + C Where C is the constant of integration. Since we are interested in the change from t = 0 to t = 60, we can calculate the definite integral from 0 to 60: ∫(from 0 to 60) (2000t + 1000) dt = [1000t^2 + 1000t] (from 0 to 60) = (1000 * 60^2 + 1000 * 60) - (1000 * 0^2 + 1000 * 0) = (1000 * 3600 + 1000 * 60) - 0 = 366000 So, the candidate will have 366,000 supporters after 60 days, assuming she had no supporters initially. |
3660000 |
| zhipu | To find out how many supporters the candidate will have after 60 days, we need to calculate the total number of votes she has gained over that period. The rate at which she is gaining votes is given by the function \( V(t) = 2000t + 1000 \), where \( V(t) \) is the number of votes gained per day and \( t \) is the number of days since she announced her candidacy. To find the total number of votes after 60 days, we need to integrate this rate over the interval from \( t = 0 \) to \( t = 60 \). The integral of \( V(t) \) with respect to \( t \) from 0 to 60 will give us the total votes gained: \[ \text{Total votes} = \int_{0}^{60} (2000t + 1000) \, dt \] Let's calculate this integral: \[ \int (2000t + 1000) \, dt = \int 2000t \, dt + \int 1000 \, dt \] Integrating each term separately, we get: \[ \int 2000t \, dt = 2000 \cdot \frac{t^2}{2} = 1000t^2 \] \[ \int 1000 \, dt = 1000t \] Now, we evaluate these from \( t = 0 \) to \( t = 60 \): \[ \left[ 1000t^2 + 1000t \right]_{0}^{60} \] Substitute \( t = 60 \) into the integrated function: \[ 1000(60)^2 + 1000(60) = 1000 \cdot 3600 + 1000 \cdot 60 \] \[ = 3600000 + 60000 \] \[ = 3660000 \] Substitute \( t = 0 \) into the integrated function: \[ 1000(0)^2 + 1000(0) = 0 \] Now, subtract the value at \( t = 0 \) from the value at \( t = 60 \): \[ 3660000 - 0 = 3660000 \] So, after 60 days, the candidate will have 3,660,000 supporters, assuming that she had no supporters at \( t = 0 \). |
3660000 |
| gpto1 | To find out how many supporters the candidate will have after 60 days, we'll need to calculate the total number of votes she gains over that period. The rate at which she is gaining votes is given by the function: \[ R(t) = 2000t + 1000 \] where: - \( R(t) \) is the rate of gaining votes per day. - \( t \) is the number of days since she announced her candidacy. Since we know the rate of change, we can find the total number of supporters by integrating the rate function over the interval from \( t = 0 \) to \( t = 60 \) days. ### Step 1: Set Up the Integral The total number of supporters, \( S \), after 60 days is: \[ S = \int_{0}^{60} R(t) \, dt = \int_{0}^{60} (2000t + 1000) \, dt \] ### Step 2: Integrate the Function We'll integrate each term of the function separately. #### Integrate \( 2000t \): \[ \int_{0}^{60} 2000t \, dt = 2000 \int_{0}^{60} t \, dt = 2000 \left[ \frac{1}{2}t^2 \right]_{0}^{60} \] #### Integrate \( 1000 \): \[ \int_{0}^{60} 1000 \, dt = 1000 \int_{0}^{60} dt = 1000 \left[ t \right]_{0}^{60} \] ### Step 3: Evaluate the Integrals #### Evaluate the first integral: \[ 2000 \left[ \frac{1}{2}t^2 \right]_{0}^{60} = 2000 \left( \frac{1}{2}(60)^2 - \frac{1}{2}(0)^2 \right) \] \[ = 2000 \left( \frac{1}{2}(3600) - 0 \right) = 2000 \left( 1800 \right) = 3,600,000 \] #### Evaluate the second integral: \[ 1000 \left[ t \right]_{0}^{60} = 1000 (60 - 0) = 1000 \times 60 = 60,000 \] ### Step 4: Add the Results \[ S = \text{First Integral} + \text{Second Integral} = 3,600,000 + 60,000 = 3,660,000 \] ### Conclusion After 60 days, the candidate will have **3,660,000 supporters**. |
3660000 |
| deepseekv3 | To determine the number of supporters the mayoral candidate will have after 60 days, we need to integrate the rate at which she is gaining votes over the 60-day period. Given: - The rate of gaining votes is \( 2000t + 1000 \) votes per day, where \( t \) is the number of days since the announcement. - The candidate had no supporters at \( t = 0 \). We want to find the total number of supporters after 60 days, which is the integral of the rate function from \( t = 0 \) to \( t = 60 \). ### Step 1: Set up the integral The total number of supporters \( S \) after 60 days is given by: \[ S = \int_{0}^{60} (2000t + 1000) \, dt \] ### Step 2: Integrate the function Integrate \( 2000t + 1000 \) with respect to \( t \): \[ \int (2000t + 1000) \, dt = 1000t^2 + 1000t + C \] where \( C \) is the constant of integration. ### Step 3: Evaluate the definite integral Evaluate the integral from \( t = 0 \) to \( t = 60 \): \[ S = \left[ 1000t^2 + 1000t \right]_{0}^{60} \] \[ S = \left( 1000(60)^2 + 1000(60) \right) - \left( 1000(0)^2 + 1000(0) \right) \] \[ S = 1000(3600) + 1000(60) \] \[ S = 3,600,000 + 60,000 \] \[ S = 3,660,000 \] ### Final Answer The candidate will have **3,660,000 supporters** after 60 days. |
3660000 |