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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)=50-23 sin(pi(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.
When 
t= seconds

Kajal holds a ruler in some wavy water. The depth of the water t t seconds after she starts measuring it, in cm \mathrm{cm} , is given by\newlineD(t)=5023sin(π(t+0.23)). D(t)=50-23 \sin (\pi(t+0.23)) . \newlineAfter she starts measuring, when is the first time the depth of the waves is at its average value? Give an exact answer.\newlineWhen t= t= \square seconds

Full solution

Q. Kajal holds a ruler in some wavy water. The depth of the water t t seconds after she starts measuring it, in cm \mathrm{cm} , is given by\newlineD(t)=5023sin(π(t+0.23)). D(t)=50-23 \sin (\pi(t+0.23)) . \newlineAfter she starts measuring, when is the first time the depth of the waves is at its average value? Give an exact answer.\newlineWhen t= t= \square seconds
  1. Average Depth Value: The average value of the depth is given by the constant term in the equation, which is 50cm50\,\text{cm}.
  2. Finding Average Depth: To find when the depth is at its average value, we set D(t)D(t) equal to 5050 cm.50=5023sin(π(t+0.23))50 = 50 - 23 \sin(\pi(t + 0.23))
  3. Isolating Trigonometric Function: Subtract 5050 from both sides to isolate the trigonometric function.\newline0=23sin(π(t+0.23))0 = -23 \sin(\pi(t + 0.23))
  4. Solving for Sine Function: Divide both sides by 23-23 to solve for the sine function.\newline0=sin(π(t+0.23))0 = \sin(\pi(t + 0.23))
  5. Finding Average Value of Sine: The sine function is zero at its average value, which occurs at multiples of π\pi. So we need to find the smallest positive tt such that π(t+0.23)\pi(t + 0.23) is a multiple of π\pi.π(t+0.23)=0\pi(t + 0.23) = 0
  6. Solving for t: Divide both sides by π\pi to solve for tt.t+0.23=0t + 0.23 = 0
  7. Final Result: Subtract 0.230.23 from both sides to find tt.\newlinet=0.23t = -0.23

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