Question

A Cepheid variable star is a star whose brightness alternately increases and decreases. For a certain star, the interval between times of maximum brightness is 3.4 days. The average brightness of this star is 4.0 and its brightness changes by plusminus 0.55. In view of these data, the brightness of the star at time t, where t is measured in days, has been modeled by the function B(t) = 4.0 + 0.55 sin(2 pi t/3.4). Find the rate of change of the brightness after t days. dB/dt = Find, correct to two decimal places, the rate of increase after three days. dB/dt =

Answer 1

The rate of change of the brightness B(t) after t days is found by differentiating the given function with respect to time, resulting in dB/dt = 0.55 • (2π/3.4) • cos(2πt/3.4). To find the specific rate after three days, substitute t = 3 into this derivative and calculate the value.

Step-by-step explanation:

To find the rate of change of the brightness of the star after t days, we need to differentiate the given brightness function B(t) = 4.0 + 0.55 sin(2 π t/3.4) with respect to time t. Using the chain rule, the derivative of the sine function is the cosine function multiplied by the inside function's derivative. So, the rate of change of brightness dB/dt is:

dB/dt = 0.55 • (2π/3.4) • cos(2πt/3.4)

To find the rate of increase after three days, we substitute t = 3 into our derivative:

dB/dt at t = 3 would be 0.55 • (2π/3.4) • cos(2π•3/3.4). After calculating this value and rounding to two decimal places, we would obtain the rate of increase in brightness after three days.