To test the resistance of a new product to changes in static pressure, the product is placed in a
controlled environment. The static pressure in this environment as a function of time can be
described by the cosine function. The maximum static pressure is 5000 pascals, the minimum is 100
pascals, and at π = π, the static pressure is at its maximum. In 40 hours the maximum static
pressure is reached 5 times (including the time at π = πhours and π = ππhours.) What is the
equation of the cosine function that describes the static pressure in the environment?
Here is what I got so far: f(x) = 2450 sin (k x) + 2550
How do I find the value of k?
From the statement:
In 40 hours the maximum static
pressure is reached 5 times
we can see that the period is 8 hours,
period = 2Ο/k
k = 2Ο/period = 2Ο/8 = Ο/4
Why did you choose a sine function, when they recommended the cosine function?
The cosine curve has a maximum when x = 0, so it would be the obvious choice:
f(x) = 2450cos((Ο/4)x) + 2550
checking:
We should have a max when x = 0
a min when x = 4
when x = 0, f(0) = 2450cos0 + 2550 = 2450(1) + 2450 = 5000 , check!
when x = 4, f(4) = 2450cos(Ο) + 2550 = 2450(-1)+2450 = 100 , check!
when x = 8 , f(8) = 2450cos(2Ο)+2550 = 5000, check
To find the value of k in the equation f(x) = 2450 sin (k x) + 2550, we can use the given information about the maximum and minimum static pressure.
The cosine function can be represented as a sine function with a phase shift of Ο/2. Since at π = π, the static pressure is at its maximum, we can write the equation as f(x) = 2450 sin (k x + Ο/2) + 2550.
The maximum static pressure is 5000 pascals. Substitute this into the equation to get:
5000 = 2450 sin (k * 0 + Ο/2) + 2550
Simplifying the equation:
5000 - 2550 = 2450 sin (Ο/2)
2450 = 2450 sin (Ο/2)
sin (Ο/2) = 1
So, the equation becomes:
2450 = 2450 * 1
This equation is true regardless of the value of k. Therefore, k can be any real number.
Hence, the value of k is variable and can be any real number.