We are told that the maximum pressure P produced from a sound with a frequency of 20 hertz is 0.02 millipascals. We want to write a sine model that gives us P as a function of the time t. To begin, we will recall the general form of a sine function.
P = a sin bt
In this equation, a and b are non-zero real numbers.
We know that an amplitude of basic sine function is 1. In this case we are given that the maximum pressure P is 0.02 millipascals. This means that the amplitude of our function is 0.02. In the general form of a sine function, a represents change of amplitude of the graph. Therefore, in our case we can set a= 0.02.
P = 0.02 sin btNow, we need to find b. We know that the frequency and the period are inversely proportional. In our case, the frequency is equal to 20 hertz and the period is 2Ď€ b. We can substitute these information into the relationship involving frequency and period.
frequency=1/period ⇒ 20=1/2π b
Let's solve this equation for b.
Now that we have a and b, we can complete the equation for the maximum pressure.
P = 0.02 sin 40Ď€t
To graph this model, we need to identify the points where x-intercepts, maximum values, and minimum values occur. To do so, we will substitute a= 0.02, b= 40Ď€ into the known formulas for the key points of a sine function. Let's show this in a table!
Formula
Substitute
Simplify
x-intercepts
(0,0),(1/2*2Ď€/b,0), (2Ď€/b,0)
(1/2*2Ď€/40Ď€,0), (2Ď€/40Ď€,0)
(0,0), (0.025,0), (0.05,0)
Maximum
(1/4* 2Ď€/b, a)
(1/4*2Ď€/40Ď€, 0.02 )
(0.0125,0.02)
Minimum
(3/4*2Ď€/b, - a )
(3/4*2Ď€/40Ď€, - 0.02)
(0.0375,- 0.02)
We can graph the function by plotting the points that we found and connecting them with a smooth curve. Let's do it!
