Question

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
( c(x)=0.000006x^{2}-0.003x + 2 ), for ( x = 350 )
the total cost is about ( $square )
(round the final answer to the nearest cent as needed. round all intermediate values to the nearest thousandth as needed)

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 \(x = 90\): \(C^{\prime}(90)=0.000006\times90^{2}-0.003\times90 + 2\)

$$
LATEXBLOCK0
$$

• For \(x = 180\): \(C^{\prime}(180)=0.000006\times180^{2}-0.003\times180 + 2\)

$$
LATEXBLOCK1
$$

Then \(S=(C^{\prime}(0)+C^{\prime}(90)+C^{\prime}(180))\times\Delta x\)

$$
LATEXBLOCK2
$$

Answer:

\(488.97\)