Kajal holds a ruler in some wavy water. The depth of the water t seconds after she starts measuring it, in cm, is given by D (t) = 50-23sin (π (t+0.23)). After she starts measuring, when is the first time the depth of the waves is at its average value? Give an exact answer.

t = 0.77

Step-by-step explanation:

D (t) = 50 - 23 sin (π (t + 0.23))

The value of the sine function has a maximum of - 1 and a minimum of 1.

The average value of the sine function is 0.

At maximum sine value:

D (t) = 50 - 23 (1) = 50 - 23 = 27

At minimum sine value:

D (t) = 50 - 23 (-1) = 50 + 23 = 73

At average sine value of 0:

D (t) = 50 - 23 (0) = 50 - 0 = 50

The average depth of the waves is 50 cm.

Now we need to find at what time, t, that occurs.

D (t) = 50 - 23 sin (π (t + 0.23)) = 50

50 - 23 sin (π (t + 0.23)) = 50

-23 sin (π (t + 0.23)) = 0

sin (π (t + 0.23)) = 0

The value of the sine is 0 at 0, π, 2π, ..., nπ

π (t + 0.23) = nπ

t + 0.23 = n

t = n - 0.23

n is all integers, but here we are concerned with the first occurrence after time equals zero, so we want n = 1.

t = 1 - 0.23

t = 0.77