Question

Caswell started studying how the number of branches on his tree grows over time. The relationship between the elapsed time [t], in years, since Caswell started studying the tree, and the number of its branches [n(t)] is modeled by the following function: [n(t) = 15 * (1.64)⁽³·⁴⁾ ]. Complete the following sentence about the rate of change in the number of branches.

Answer 1

The rate of change in the number of branches is determined by differentiating the exponential function n(t), indicating a constant growth factor applied to the tree's branches over time.

Step-by-step explanation:

To determine the rate of change in the number of branches of Caswell's tree modeled by n(t) = 15 * (1.64)3.4t, one would differentiate the function with respect to time (t). This calculation involves exponential growth and the rules for differentiation of exponential functions. Let's take a closer look at how this is done:

• Step 1: Start by writing down the function n(t) = 15 * (1.64)3.4t.
• Step 2: Differentiate the function with respect to t, using the chain rule. This yields dn(t)/dt = 15 * (1.64)3.4t * ln(1.64) * 3.4.
• Step 3: The derivative dn(t)/dt represents the rate of change of the number of branches with respect to time. In this case, it shows that the rate of change is proportional to the current number of branches, multiplied by the constant factor ln(1.64) * 3.4.

The constant rate of growth corresponds to the base of the exponential function (here, 1.64) and the exponent's coefficient (3.4 in this case), which can be understood as a growth factor applied to the initial quantity over time.