The displacement (in meters) of a wave is given according to y =0.26 sin `(pit- 3.7pix)`, where t is in seconds and x is in meters. What is the displacement y when t =38 s and x= 13 m?

Options

A. `-0.14m`
B. `-0.080m`
C. `-0.041 m`
D. `-0.019m`

Answer

To find the displacement \( y \) when \( t = 38 \) seconds and \( x = 13 \) meters, we start with the wave equation given:

\[ y = 0.26 \sin(\pi t - 3.7 \pi x) \]

### Step 1: Substitute the values of \( t \) and \( x \)

We need to substitute \( t = 38 \) seconds and \( x = 13 \) meters into the equation:

\[ y = 0.26 \sin(\pi \cdot 38 - 3.7 \pi \cdot 13) \]

### Step 2: Calculate \( \pi t \) and \( 3.7 \pi x \)

Now, calculate each term separately:

1. Calculate \( \pi \cdot 38 \):
\[ \pi \cdot 38 \approx 3.14 \cdot 38 \approx 119.38 \text{ (approximately)} \]

2. Calculate \( 3.7 \pi \cdot 13 \):
\[ 3.7 \cdot \pi \cdot 13 \approx 3.7 \cdot 3.14 \cdot 13 \approx 152.46 \text{ (approximately)} \]

### Step 3: Substitute back into the sine function

Now substitute these values back into the sine function:

\[ y = 0.26 \sin(119.38 - 152.46) \]

### Step 4: Calculate the argument of the sine function

Now calculate the argument of the sine function:

\[ 119.38 - 152.46 \approx -33.08 \]

### Step 5: Calculate \( \sin(-33.08) \)

Now we need to find the sine of \(-33.08\) degrees. Using a calculator or sine table:

\[ \sin(-33.08) \approx -0.544 \]

### Step 6: Calculate the displacement \( y \)

Now substitute this value back into the equation for \( y \):

\[ y = 0.26 \cdot (-0.544) \approx -0.14144 \]

### Final Answer

Thus, the displacement \( y \) when \( t = 38 \) seconds and \( x = 13 \) meters is approximately:

\[ y \approx -0.141 \text{ meters} \]

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