Our cycle of normal breathing takes place every 5 seconds. Velocity of air flow, y, measured in liters per second, after x seconds is modeled by : y=0.6sin(2pi/5)x. Velocity of air flow is positive when we inhale.
Within each breathing cycle, when are we inhaling 0.3 liters per second? Round to the nearest tenth of a second.
To find when the velocity of air flow is 0.3 liters per second during inhalation, we need to solve the equation y = 0.3 for x.
The given equation is: y = 0.6sin((2π/5)x)
Setting y = 0.3, we have:
0.3 = 0.6sin((2π/5)x)
Divide both sides by 0.6:
0.3 / 0.6 = sin((2π/5)x)
Simplify:
0.5 = sin((2π/5)x)
To find the values of x, we need to take the inverse sine (also known as arcsin or sin^(-1)) of both sides:
arcsin(0.5) = arcsin(sin((2π/5)x))
The arcsin of 0.5 is 30 degrees or π/6 radians. However, we should note that the range of arcsin is only from -90 degrees to 90 degrees or -π/2 to π/2.
Therefore, we need to find the reference angle in the first or second quadrant that has a sine value of 0.5, which is 30 degrees or π/6 radians.
We need to solve the equation (2π/5)x = π/6 for x:
2π/5 * x = π/6
Multiply both sides by 5/2:
x = (π/6) * (5/2)
x = (5π/12)
To find the time in seconds, we multiply x by 5 since the whole breathing cycle takes 5 seconds.
Time = (5π/12) * 5 = (25π/12) seconds
To round to the nearest tenth of a second, we get:
Time ≈ 6.5 seconds
Therefore, we are inhaling at a rate of 0.3 liters per second approximately 6.5 seconds into each breathing cycle.
To find out when we are inhaling at a rate of 0.3 liters per second, we need to solve the equation y = 0.3, where y represents the velocity of air flow.
The equation given to model the velocity of air flow is y = 0.6sin(2π/5x).
Setting 0.6sin(2π/5x) equal to 0.3, we have:
0.6sin(2π/5x) = 0.3.
Now, we can solve for x:
sin(2π/5x) = 0.3/0.6.
sin(2π/5x) = 0.5.
To find the value of x, we need to take the inverse sine (arcsin) of both sides:
2π/5x = arcsin(0.5).
Using a calculator, we find that the arcsin(0.5) is approximately 0.5236 radians.
Now, we can solve for x:
2π/5x = 0.5236.
Dividing both sides by 2π/5, we get:
x = 0.5236 / (2π/5).
Calculating this value, x is approximately 2.618 seconds.
However, this only gives us one solution within one breathing cycle. To find out how often we inhale at 0.3 liters per second, we need to consider the periodicity of the sinusoidal function.
The general formula for the period of a sine function is T = 2π/b, where b is the coefficient of x in the sine function.
In our case, the coefficient of x is (2π/5), so the period is T = 2π / (2π/5) = 5 seconds.
Since the velocity of air flow repeats every 5 seconds, we need to find all the solutions for x within one breathing cycle (5 seconds).
So, the final step is to find all the x values within the range 0 to 5 seconds when y = 0.3.
We can set up an equation to find these values:
0.6sin(2π/5x) = 0.3.
Now, solve for x by finding all the values between 0 and 5 where y = 0.3.
Using a graphing calculator or a table of values, we find that the solutions within one breathing cycle are approximately:
x ≈ 1.256, 3.142, and 4.028 seconds.
Rounding these values to the nearest tenth of a second, we have:
x ≈ 1.3, 3.1, and 4.0 seconds.
Therefore, within each breathing cycle, we are inhaling at a rate of 0.3 liters per second at approximately 1.3, 3.1, and 4.0 seconds (rounded to the nearest tenth of a second).
.3=.6sin(2pi x/5)
solve for x.