An oil refinery produces oil at a variable rate given by the following equation, where \( t \) is measured in days and \( Q \) is measure in barrels, \[ Q^{\prime}(t)=\left\{\begin{array}{ll}600 & \text { if } 0 \leq t \leq 30 \\ 2500=40 t & \text { if } 30 s t \leqslant 40 \\ 400 & \text { if } t \geq 40\end{array}\right. \] 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 barels produced over the interval \( [70,90] \),

In the first 35 days, the production rate is constant at 600 barrels per day. Therefore, for the first 30 days, we have: \[ Q(30) = 600 \times 30 = 18000 \text{ barrels} \] From day 30 to day 35, the production rate switches to \( 2500 - 40t \). When \( t = 30 \), the production is actually \( 2500 - 40(30) = 2500 - 1200 = 1300 \) barrels for t=30. For 5 days at that rate, \[ Q(35) = 1300 \times 5 = 6500 \text{ barrels} \] Total production for the first 35 days = \[ 18000 + 6500 = 24500 \text{ barrels} \] Now for the first 50 days, we calculate as follows: - Days 0 to 30 = 18000 barrels - Days 30 to 40: From day 30 to day 40 (10 days) at \( 1500 \) barrels (calculated from the new rate), - 1500 barrels per day over 10 days = \( 1500 \times 10 = 15000 \text{ barrels} \) - Days 40 to 50: Production returns to 400 barrels a day for 10 days, which gives us \( 400 \times 10 = 4000 \text{ barrels} \). Thus, total for the first 50 days = \[ 18000 + 15000 + 4000 = 37000 \text{ barrels} \] For the interval [70, 90], we know that the production remains at 400 barrels per day since \( t \geq 40 \). Thus, to determine the output over these 20 days: \[ 400 \times 20 = 8000 \text{ barrels} \] Therefore, the conclusion is: - Total in the first 35 days: 24500 barrels - Total in the first 50 days: 37000 barrels - Production in the interval [70, 90]: 8000 barrels.