For how many integral values of x `, sinphi=((3x-2))/(4)` , where `0^(@)lephile90^(@)`

Options

A. 2
B. 3
C. 0
D. 1

Answer

To solve the problem, we need to find the integral values of \( x \) such that:

\[ \sin \phi = \frac{3x - 2}{4} \]

where \( 0^\circ < \phi < 90^\circ \).

### Step 1: Understand the range of sine function
The sine function, \( \sin \phi \), for \( 0^\circ < \phi < 90^\circ \) takes values in the range \( (0, 1) \). This means:

\[ 0 < \sin \phi < 1 \]

### Step 2: Set up the inequality
From the equation \( \sin \phi = \frac{3x - 2}{4} \), we can set up the following inequality based on the range of sine:

\[ 0 < \frac{3x - 2}{4} < 1 \]

### Step 3: Solve the inequalities
Now, we will solve the two parts of the inequality separately.

1. **First part: \( \frac{3x - 2}{4} > 0 \)**

Multiply both sides by 4:

\[ 3x - 2 > 0 \]

Adding 2 to both sides:

\[ 3x > 2 \]

Dividing by 3:

\[ x > \frac{2}{3} \]

2. **Second part: \( \frac{3x - 2}{4} < 1 \)**

Multiply both sides by 4:

\[ 3x - 2 < 4 \]

Adding 2 to both sides:

\[ 3x < 6 \]

Dividing by 3:

\[ x < 2 \]

### Step 4: Combine the inequalities
Now we combine the results from both parts:

\[ \frac{2}{3} < x < 2 \]

### Step 5: Determine integral values
The integral values of \( x \) that satisfy this inequality are:

- The smallest integer greater than \( \frac{2}{3} \) is 1.
- The largest integer less than 2 is 1.

Thus, the only integral value of \( x \) that satisfies the inequality is:

\[ x = 1 \]

### Conclusion
Therefore, there is only **1 integral value** of \( x \) that satisfies the equation.