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Part 1 of 3 \( Q^{\prime}(t)=\left\{\begin{array}{ll}600 & \text { if } 0 \leq t<30 \\ 2900-70 \mathrm{t} & \text { if } 30 \leq t<40 \\ 400 & \text { if } t \geq 40\end{array}\right. \) a. How many barrels are produced in the first 35 days? b. How many barrels are produced in the first 50 days? c. Without using calculus, determine the number of barrels produced over the interval \( [70,80] \). \( 53 \% \)

Ask by Carter Lowe. in the United States
Jan 23,2025

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Answer

a. 21,125 barrels b. 26,500 barrels c. 4,000 barrels

Solution

Let's analyze the production function and solve each part step by step. Given the rate of production \( Q'(t) \) as a piecewise function: \[ Q^{\prime}(t)= \begin{cases} 600 & \text{if } 0 \leq t < 30 \\ 2900 - 70t & \text{if } 30 \leq t < 40 \\ 400 & \text{if } t \geq 40 \end{cases} \] Here, \( t \) represents time in days, and \( Q'(t) \) is the rate of production in barrels per day. To find the total number of barrels produced over a specific interval, we'll integrate \( Q'(t) \) over that interval. ### **a. Number of Barrels Produced in the First 35 Days** **Approach:** 1. **From Day 0 to Day 30:** The production rate is constant at 600 barrels/day. 2. **From Day 30 to Day 35:** The production rate decreases linearly as \( 2900 - 70t \). **Calculations:** 1. **Production from Day 0 to Day 30:** \[ \text{Total} = 600 \, \text{barrels/day} \times 30 \, \text{days} = 18,000 \, \text{barrels} \] 2. **Production from Day 30 to Day 35:** \[ \text{Total} = \int_{30}^{35} (2900 - 70t) \, dt \] \[ = \left[2900t - 35t^2\right]_{30}^{35} \] \[ = (2900 \times 35 - 35 \times 35^2) - (2900 \times 30 - 35 \times 30^2) \] \[ = (101,500 - 42,875) - (87,000 - 31,500) \] \[ = 58,625 - 55,500 = 3,125 \, \text{barrels} \] 3. **Total Production in 35 Days:** \[ 18,000 \, \text{barrels} + 3,125 \, \text{barrels} = 21,125 \, \text{barrels} \] **Answer:** \[ \boxed{21,\!125 \text{ barrels}} \] ### **b. Number of Barrels Produced in the First 50 Days** **Approach:** 1. **From Day 0 to Day 30:** Production rate is 600 barrels/day. 2. **From Day 30 to Day 40:** Production rate decreases as \( 2900 - 70t \). 3. **From Day 40 to Day 50:** Production rate stabilizes at 400 barrels/day. **Calculations:** 1. **Production from Day 0 to Day 30:** \[ 600 \times 30 = 18,000 \, \text{barrels} \] 2. **Production from Day 30 to Day 40:** \[ \text{Total} = \int_{30}^{40} (2900 - 70t) \, dt \] \[ = \left[2900t - 35t^2\right]_{30}^{40} \] \[ = (116,000 - 56,000) - (87,000 - 31,500) \] \[ = 60,000 - 55,500 = 4,500 \, \text{barrels} \] 3. **Production from Day 40 to Day 50:** \[ 400 \times 10 = 4,000 \, \text{barrels} \] 4. **Total Production in 50 Days:** \[ 18,000 + 4,500 + 4,000 = 26,500 \, \text{barrels} \] **Answer:** \[ \boxed{26,\!500 \text{ barrels}} \] ### **c. Number of Barrels Produced Over the Interval [70, 80]** **Approach:** - **From Day 70 to Day 80:** Since \( t \geq 40 \), the production rate is constant at 400 barrels/day. **Calculations:** \[ \text{Total} = 400 \, \text{barrels/day} \times 10 \, \text{days} = 4,000 \, \text{barrels} \] **Answer:** \[ \boxed{4,\!000 \text{ barrels}} \]

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In the first 35 days, you can calculate production by adding the barrels produced in the given time intervals. From day 0 to day 30, you have a constant production of 600 barrels/day. That's \(600 \times 30 = 18,000\) barrels. From day 30 to day 35, use the formula \(2900 - 70t\). Plugging in \(t=30\) gives you a rate of 2900 - 2100 = 800 barrels/day. Over 5 days, that’s \(800 \times 5 = 4,000\) barrels. Thus, the total for 35 days is \(18,000 + 4,000 = 22,000\) barrels. For the first 50 days, you'll follow similar steps. Days 0 to 30 again yield 18,000 barrels. From day 30 to day 40 (the next 10 days), use the same formula. For \(t=30\) to \(t=40\), the production decreases from 800 to 0. The average production during this segment is \((800 + 0) / 2 = 400\) barrels/day over 10 days gives \(400 \times 10 = 4,000\) barrels. From day 40 to day 50, you have 400 barrels/day for 10 days resulting in \(400 \times 10 = 4,000\) barrels. The total for 50 days then is \(18,000 + 4,000 + 4,000 = 26,000\) barrels. To determine the production from day 70 to day 80, note that \(t\) is greater than 40, so the production rate is consistent at 400 barrels/day. For this 10-day interval, that's \(400 \times 10 = 4,000\) barrels. Voila! You've got your answers without a hitch!

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