A conduction heat wave is caused to pass through a metal bar of average temperature Tmean = 38.0 ¢XC, altering its temperature sinusoidally with an amplitude of Ti = 3.00 ¢XC. The equation which gives the temperature, T(x,t), of the metal bar at any place x (in meters) inside it at any time t (in seconds) is:
T(x,t) = 38 + 3 sin[2£k(0.0290t ¡V5.00x) + 1.5pie]
What is the value of T(x,t) when t = 220.0 s, and x = 0.25m?
I first worked out the wavelength = 0.2m and then the period = 34.5 seconds then i rewrote the equation to be T(x,t) = 38+3cos[2n(0.029*220/34.5 - 5*0.25/0.2) + 1.5pie] which gave me 35.5 which is not the correct answer of 35.9. What have i done wrong?
It seems like you made a calculation error in your equation. Let's go through the steps again to find the correct value of T(x,t).
Given equation: T(x,t) = 38 + 3 sin[2πk(0.0290t - 5.00x) + 1.5π]
We are looking for the value of T(x,t) when t = 220.0 s and x = 0.25 m.
Step 1: Calculate the wavelength (λ):
In the given equation, the coefficient of x is -5.00. Since the coefficient of x represents the number of wavelengths that fit in the given distance, we can calculate the wavelength as the reciprocal of the coefficient of x.
λ = 1 / |(-5.00)| = 0.2 m
Step 2: Calculate the period (T):
The period of the sinusoidal function can be calculated as the time it takes to complete one full cycle.
T = λ/v, where v is the speed of the wave. In this case, v = 0.0290 m/s.
T = 0.2 m / 0.0290 m/s = 6.897 s (approx)
Step 3: Plug in the values into the equation:
We have t = 220.0 s and x = 0.25 m.
T(x,t) = 38 + 3 sin[2π(0.0290)(220 - 5.00*0.25) + 1.5π]
T(x,t) = 38 + 3 sin[2π(6.45) + 1.5π]
T(x,t) = 38 + 3 sin[12.859π + 1.5π]
T(x,t) = 38 + 3 sin(40.91π)
Step 4: Convert radians to degrees:
To use the calculator, we need to convert radians to degrees.
40.91π radians ≈ 128.56°
Step 5: Calculate the value of sin(128.56°):
Using a scientific calculator or the trigonometric functions in your calculator, evaluate the sine of 128.56°.
sin(128.56°) ≈ 0.477
Step 6: Substitute the calculated value back into the equation:
T(x,t) = 38 + 3(0.477)
T(x,t) = 38 + 1.431
T(x,t) ≈ 39.431
Therefore, the value of T(x,t) when t = 220.0 s and x = 0.25 m is approximately 39.431 ¢XC