Total number of solutions of ` sinx=(|x|)/(10)` is equal to

Options

A. 4
B. 6
C. 7
D. None of these

Answer

To find the total number of solutions for the equation \( \sin x = \frac{|x|}{10} \), we can follow these steps:

### Step 1: Understand the Functions
We have two functions to analyze:
1. \( y = \sin x \)
2. \( y = \frac{|x|}{10} \)

### Step 2: Analyze the Sine Function
The sine function oscillates between -1 and 1. Therefore, the maximum and minimum values of \( \sin x \) are:
- Maximum: \( 1 \)
- Minimum: \( -1 \)

### Step 3: Scale the Sine Function
Since we have \( \sin x = \frac{|x|}{10} \), we can rewrite this as:
\[ |x| = 10 \sin x \]
This means the maximum value of \( |x| \) can be \( 10 \) (when \( \sin x = 1 \)) and the minimum value can be \( -10 \) (when \( \sin x = -1 \)).

### Step 4: Graph the Functions
- The graph of \( y = \sin x \) oscillates between -1 and 1.
- The graph of \( y = \frac{|x|}{10} \) is a V-shaped graph that opens upwards with its vertex at the origin (0,0) and has slopes of \( \frac{1}{10} \) on both sides.

### Step 5: Determine Intersection Points
To find the total number of solutions, we need to determine how many times these two graphs intersect:
- The sine function oscillates and crosses the x-axis infinitely many times.
- The line \( y = \frac{|x|}{10} \) will intersect the sine wave.

### Step 6: Count the Intersections
1. For \( x \geq 0 \):
- The line \( y = \frac{x}{10} \) will intersect the sine curve multiple times.
- The sine function reaches its maximum of 1 at \( x = \frac{\pi}{2} \) and returns to 0 at \( x = \pi \), then goes negative until \( x = 3\pi/2 \), and returns to 0 at \( x = 2\pi \), continuing this pattern.
- Each cycle of \( 2\pi \) will have two intersections in the first half (0 to \( \pi \)) and two in the second half (\( \pi \) to \( 2\pi \)).
- Thus, in each interval of \( 2\pi \), there are 2 intersections.

2. For \( x < 0 \):
- The line \( y = \frac{-x}{10} \) will also intersect the sine curve similarly.
- The number of intersections will mirror those found in the positive side.

### Step 7: Total Number of Solutions
- In each interval of \( 2\pi \), we have 2 intersections for \( x \geq 0 \) and 2 for \( x < 0 \).
- Therefore, the total number of solutions is \( 2 + 2 = 4 \) for each cycle of \( 2\pi \).
- Since the sine function is periodic, we can find that there are a total of 6 intersections within the range of interest.

### Final Answer
The total number of solutions to the equation \( \sin x = \frac{|x|}{10} \) is **6**.

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