One-fourth of a herd of cows is in the forest. Twice the square root of the herd has gone to mountains and he remaining 15 are on the banks of a river. The total number of cows is

Options

A. 6
B. 100
C. 63
D. 36

Answer

To solve the problem, we will define the total number of cows in the herd as \( x \).

### Step 1: Set up the equation based on the information given.

According to the problem:
- One-fourth of the herd is in the forest, which is \( \frac{x}{4} \).
- Twice the square root of the herd has gone to the mountains, which is \( 2\sqrt{x} \).
- The remaining cows are 15, which means after accounting for the cows in the forest and the mountains, 15 cows are left.

So, we can write the equation as:
\[ \frac{x}{4} + 2\sqrt{x} + 15 = x \]

### Step 2: Rearrange the equation.

To simplify the equation, we will move all terms to one side:
\[ \frac{x}{4} + 2\sqrt{x} + 15 - x = 0 \]

This can be rewritten as:
\[ -\frac{3x}{4} + 2\sqrt{x} + 15 = 0 \]

Multiplying the entire equation by -4 to eliminate the fraction gives:
\[ 3x - 8\sqrt{x} - 60 = 0 \]

### Step 3: Substitute \( \sqrt{x} \) with a new variable.

Let \( y = \sqrt{x} \). Then, \( x = y^2 \). Substituting this into the equation gives:
\[ 3y^2 - 8y - 60 = 0 \]

### Step 4: Solve the quadratic equation.

Now we can solve the quadratic equation \( 3y^2 - 8y - 60 = 0 \) using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 3 \), \( b = -8 \), and \( c = -60 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-8)^2 - 4 \cdot 3 \cdot (-60) = 64 + 720 = 784 \]

Now substituting into the quadratic formula:
\[ y = \frac{8 \pm \sqrt{784}}{2 \cdot 3} = \frac{8 \pm 28}{6} \]

Calculating the two possible values for \( y \):
1. \( y = \frac{36}{6} = 6 \)
2. \( y = \frac{-20}{6} = -\frac{10}{3} \) (not valid since \( y \) must be non-negative)

Thus, \( y = 6 \).

### Step 5: Find \( x \).

Now, substituting back to find \( x \):
\[ \sqrt{x} = 6 \implies x = 6^2 = 36 \]

### Conclusion

The total number of cows in the herd is \( \boxed{36} \).