a juice company has found that the marginal cost of producing x pints of fresh - squeezed orange juice is…

a juice company has found that the marginal cost of producing x pints of fresh - squeezed orange juice is given by the function below, where ( c(x) ) is in dollars. approximate the total cost of producing 270 pt of juice, using 3 subintervals over ( 0,270 ) and the left endpoint of each subinterval\n( c(x)=0.000006x^{2}-0.003x + 2 ), for ( x = 350 )\nthe total cost is about ( $square )\n(round the final answer to the nearest cent as needed. round all intermediate values to the nearest thousandth as needed)

a juice company has found that the marginal cost of producing x pints of fresh - squeezed orange juice is given by the function below, where ( c(x) ) is in dollars. approximate the total cost of producing 270 pt of juice, using 3 subintervals over ( 0,270 ) and the left endpoint of each subinterval\n( c(x)=0.000006x^{2}-0.003x + 2 ), for ( x = 350 )\nthe total cost is about ( $square )\n(round the final answer to the nearest cent as needed. round all intermediate values to the nearest thousandth as needed)

Answer

Explanation:

Step1: Calculate the width of sub - intervals

The interval is ([0,270]) and (n = 3). The width (\Delta x=\frac{270 - 0}{3}=90).

Step2: Determine the left - endpoints

The sub - intervals are ([0,90]), ([90,180]), ([180,270]). The left - endpoints are (x_0 = 0), (x_1=90), (x_2 = 180).

Step3: Calculate the sum using the left - endpoint Riemann sum formula (S=\sum_{i = 0}^{n-1}C^{\prime}(x_i)\Delta x)

  • For (x = 0): (C^{\prime}(0)=0.000006\times0^{2}-0.003\times0 + 2=2)
  • For (x = 90): (C^{\prime}(90)=0.000006\times90^{2}-0.003\times90 + 2) [ \begin{align*} C^{\prime}(90)&=0.000006\times8100-0.27 + 2\ &=0.0486-0.27+2\ &=1.7786 \end{align*} ]
  • For (x = 180): (C^{\prime}(180)=0.000006\times180^{2}-0.003\times180 + 2) [ \begin{align*} C^{\prime}(180)&=0.000006\times32400-0.54+2\ &=0.1944-0.54 + 2\ &=1.6544 \end{align*} ] Then (S=(C^{\prime}(0)+C^{\prime}(90)+C^{\prime}(180))\times\Delta x) [ \begin{align*} S&=(2 + 1.7786+1.6544)\times90\ &=(5.433)\times90\ &=488.97 \end{align*} ]

Answer:

(488.97)