The arithmetic mean of two numbers is `18(3)/(4)` and the positive square root of their product is 15. The larger of the two numbers is

Options

A. 24
B. 25
C. 20
D. 30

Answer

To solve the problem, we need to find the larger of two numbers given their arithmetic mean and the positive square root of their product. Let's break this down step by step.

### Step 1: Understand the given information

We are given:
1. The arithmetic mean of two numbers \( a \) and \( b \) is \( 18 \frac{3}{4} \).
2. The positive square root of their product \( \sqrt{ab} = 15 \).

### Step 2: Convert the arithmetic mean to an improper fraction

The arithmetic mean can be converted as follows:
\[ 18 \frac{3}{4} = 18 + \frac{3}{4} = \frac{72}{4} + \frac{3}{4} = \frac{75}{4} \]

### Step 3: Set up the equations

From the arithmetic mean, we have:
\[ \frac{a + b}{2} = \frac{75}{4} \]
Multiplying both sides by 2 gives:
\[ a + b = \frac{75}{2} \]

From the product, we have:
\[ \sqrt{ab} = 15 \implies ab = 15^2 = 225 \]

### Step 4: Use the equations to form a quadratic equation

We can express \( a \) and \( b \) as the roots of the quadratic equation:
\[ x^2 - (a + b)x + ab = 0 \]
Substituting the values we found:
\[ x^2 - \left(\frac{75}{2}\right)x + 225 = 0 \]

### Step 5: Multiply through by 4 to eliminate the fraction

To simplify, multiply the entire equation by 4:
\[ 4x^2 - 150x + 900 = 0 \]

### Step 6: Use the quadratic formula

The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 4 \), \( b = -150 \), and \( c = 900 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-150)^2 - 4 \cdot 4 \cdot 900 = 22500 - 14400 = 8100 \]
Now, applying the quadratic formula:
\[ x = \frac{150 \pm \sqrt{8100}}{8} \]
Calculating \( \sqrt{8100} = 90 \):
\[ x = \frac{150 \pm 90}{8} \]

### Step 7: Find the two possible values for \( x \)

Calculating the two possible roots:
1. \( x = \frac{150 + 90}{8} = \frac{240}{8} = 30 \)
2. \( x = \frac{150 - 90}{8} = \frac{60}{8} = 7.5 \)

### Step 8: Identify the larger number

The two numbers are \( 30 \) and \( 7.5 \). Thus, the larger of the two numbers is:
\[ \boxed{30} \]