Question
(a) Topic-2.2 Defining the Derivative of a Function and Using Derivative Notation
(b) Topic-5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
(c) Topic-7.6 Finding General Solutions Using Separation of Variables
At the beginning of 2010, a landfill contained 1400 tons of solid waste. The increasing function W models the total amount of solid waste stored at the landfill. Planners estimate that W will satisfy the differential equation \(\frac{dW}{dt}=\frac{1}{25}(W-300)\) for the next 20 years. W is measured in tons, and t is measured in years from the start of 2010.
(a) Use the line tangent to the graph of W at t = 0 to approximate the amount of solid waste that the landfill contains at the end of the first 3 months of 2010 \(\left ( time t = \frac{1}{4} \right )\).
(b) Find \(\frac{d^{2}W}{dt^{2}}\) in terms of W. Use \(\frac{d^{2}W}{dt^{2}}\) to determine whether your answer in part (a) is an underestimate or an overestimate of the amount of solid waste that the landfill contains at time \(t = \frac{1}{4}\).
(c) Find the particular solution W = W(t) to the differential equation \(\frac{dW}{dt}=\frac{1}{25}(W-300)\) with initial condition W(0) = 1400.
▶️Answer/Explanation
\(
\textbf{5(a) } \frac{dW}{dt}\bigg|_{t=0} = \frac{1}{25}(W(0) – 300) = \frac{1}{25}(1400 – 300) = 44
\)
\(
\text{The tangent line is } y = 1400 + 44t.
\)
\(
W\left(\frac{1}{4}\right) \approx 1400 + 44\left(\frac{1}{4}\right) = 1411 \text{ tons.}
\)
\(
\textbf{5(b) } \frac{d^2W}{dt^2} = \frac{1}{25} \frac{dW}{dt} = \frac{1}{625}(W – 300) \text{ and } W \geq 1400.
\)
\(
\text{Therefore } \frac{d^2W}{dt^2} > 0 \text{ on the interval } 0 \leq t \leq \frac{1}{4}.
\)
\(
\text{The answer in part (a) is an underestimate.}
\)
\(
\textbf{5(c) } \frac{dW}{dt} = \frac{1}{25}(W – 300)
\)
\(
\int \frac{1}{W – 300} \, dW = \int \frac{1}{25} \, dt
\)
\(
\ln|W – 300| = \frac{1}{25}t + C
\)
\(
\ln(1400 – 300) = \frac{1}{25}(0) + C \implies \ln(1100) = C
\)
\(
W – 300 = 1100e^{\frac{1}{25}t}
\)
\(
W(t) = 300 + 1100e^{\frac{1}{25}t}, \quad 0 \leq t \leq 20.
\)