Based on this model, use a linear approximation to the graph of k(t) at t = 2 to estimate the quantity of the substance at t = 2.1.

Question

The quantity of a substance can be modeled by the function k(t) that satisfies the differential equation dk/ dt =-1/90 (k-450). One point on this function is k(2) = 900. Based on this model, use a linear approximation to the graph of k(t) at t = 2 to estimate the quantity of the substance at t = 2.1.

Answer 1

We don't need to solve this separable differential equation, though we could easily enough, because we are asked only to use the tangent line to get an approximation of the value of the function at a specific point. This means instead all we need is the point of tangency (in this case (2 , 900)), the t-value at which to approximate the function (in this case t = 2.1), and the slope of the tangent at the point of tangency (which we will calculate in this case by simply plugging in 900 for k into the equation for dk/dt):

dk/dt | _{t = 900} = - 5. Tangent eqn: k - 900 = - 5 (t - 2). For t = 2.1, k - 900 = -.5 , k = 899.5 (g or whatever units are given).