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 \( [70,80] \). \( 53 \% \)

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:** \[ \boxed{4,\!000 \text{ barrels}} \]