Problem 36 - Calculus Early Transcendentals

Question

The owners of an oil reserve begin extracting oil at time \(t=0 .\) Based on estimates of the reserves, suppose the projected extraction rate is given by \(Q^{\prime}(t)=3 t^{2}(40-t)^{2}\) where \(0 \leq t \leq 40, Q\) is measured in millions of barrels, and \(t\) is measured in years. a. When does the peak extraction rate occur? b. How much oil is extracted in the first \(10,20,\) and 30 years? c. What is the total amount of oil extracted in 40 years? d. Is one-fourth of the total oil extracted in the first one-fourth of the extraction period? Explain.

Step-by-Step Solution

Answer

Calculate the amount of oil extracted in the first 10, 20, and 30 years, as well as the total amount of oil extracted in 40 years. Determine if one-fourth of the total oil extracted is taken in the first one-fourth of the extraction period. Answer: The peak extraction rate occurs at 20 years. To find the amount of oil extracted in the first 10, 20, and 30 years, and the total amount of oil extracted in 40 years, we need to evaluate the integrals \(Q(10)\), \(Q(20)\), \(Q(30)\), and \(Q(40)\), respectively, through numerical integration. To determine if one-fourth of the total oil extracted is taken in the first one-fourth of the extraction period, we need to compare \(Q(10)\) to \(\frac{1}{4}Q(40)\) and see if they are equal or not.

Step 1: Derive the extraction rate function

We are given the function \(Q^{\prime}(t)=3t^2(40-t)^2\). To find the maximum extraction rate, we need to find the critical points. We find the critical points by taking the derivative of the extraction rate function with respect to \(t\): \(Q^{\prime\prime}(t)=\frac{d}{dt}(3t^{2}(40-t)^{2})\) The derivative will give us information about the rate of change of the extraction rate. If \(Q^{\prime\prime}(t)=0\), it will imply that the extraction rate is either at its peak or its minimum.

Step 2: Solve for the critical points

Since \(Q^{\prime}(t)\) is a product of two functions, \(u(t)=3t^2\) and \(v(t)=(40-t)^2\), we will use the product rule to find the derivative. The product rule states that: \((u\cdot v)^{\prime}(t)=u^{\prime}(t)v(t)+u(t)v^{\prime}(t)\) Differentiate \(u(t)\) and \(v(t)\) with respect to \(t\): \(u^{\prime}(t)=\frac{d}{dt}(3t^{2})=6t\) \(v^{\prime}(t)=\frac{d}{dt}((40-t)^{2})=-2(40-t)\) Now, apply the product rule: \(Q^{\prime\prime}(t)=u^{\prime}(t)v(t)+u(t)v^{\prime}(t)=6t(40-t)^2+3t^{2}(-2)(40-t)\) Set \(Q^{\prime\prime}(t)=0\) and solve for \(t\): \(6t(40-t)^2-6t^{2}(40-t)=0\) Factor \(t\) from the equation: \(t[6(40-t)^2-6t(40-t)]=0\) The critical points are \(t=0\) and \(t=20\).

Step 3: Determine the peak extraction rate

We can now conclude that the peak extraction rate occurs when \(t=20\) years because \(t=0\) is the starting point of the problem, and we are only interested in the subsequent peak_extraction rate. The peak extraction rate occurs at 20 years. #b. Oil extraction in first 10, 20, and 30 years# To find the amount of oil extracted, we need to integrate the extraction rate function over the time intervals: \([0,10]\), \([0,20]\), and \([0,30]\).

Step 1: Integrate the extraction rate function

The extraction rate function is \(Q^{\prime}(t)=3t^{2}(40-t)^{2}\). To find the amount of oil extracted, we integrate this function with respect to time: \(Q(t)=\int_{0}^{t}3t^{2}(40-t)^{2}dt\)

Step 2: Oil extraction in first 10 years

Integrate \(Q^{\prime}(t)\) from \(0\) to \(10\) to find the amount of oil extracted in the first 10 years: \(Q(10)=\int_{0}^{10}3t^{2}(40-t)^{2}dt\)

Step 3: Oil extraction in the first 20 years

Integrate \(Q^{\prime}(t)\) from \(0\) to \(20\) to find the amount of oil extracted in the first 20 years: \(Q(20)=\int_{0}^{20}3t^{2}(40-t)^{2}dt\)

Step 4: Oil extraction in the first 30 years

Integrate \(Q^{\prime}(t)\) from \(0\) to \(30\) to find the amount of oil extracted in the first 30 years: \(Q(30)=\int_{0}^{30}3t^{2}(40-t)^{2}dt\) We need to evaluate the integrals numerically to find how much oil is extracted over each of these time intervals. #c. Total oil extraction in 40 years#

Step 1: Oil extraction in 40 years

Integrate \(Q^{\prime}(t)\) from \(0\) to \(40\) to find the total amount of oil extracted in 40 years: \(Q(40)=\int_{0}^{40}3t^{2}(40-t)^{2}dt\) Find the value of this integral numerically. #d. Is one-fourth of the total oil extracted in the first one-fourth of the extraction period?#

Step 1: Calculate one-fourth of the extraction period

Divide the \(40\) years by \(4\) to get one-fourth of the extraction period: One-fourth of the extraction period is \(40/4=10\) years.

Step 2: Compare oil extraction

Compare the oil extracted in the first 10 years to one-fourth of the total oil extracted in 40 years: \(Q(10)\) vs. \(\frac{1}{4}Q(40)\) Determine if they are equal or not to see if one-fourth of the total oil is extracted in the first 10 years. - If they are equal, then one-fourth of the total oil is extracted in the first one-fourth of the extraction period. - If they are not equal, then one-fourth of the total oil is not extracted in the first one-fourth of the extraction period.