The product of two numbers is 48. The squares is 100. Sum of these two numbers is equal to :

Question

Accepted Answer

To solve the problem, we need to find two numbers whose product is 48 and whose squares sum to 100. We are also given that the sum of these two numbers is equal to one of the options provided, specifically Option 1, which is 14.

Let's denote the two numbers as $ x $ and $ y $.

1. **Product of the numbers:**
   $$
   x \times y = 48
   $$

2. **Sum of the squares of the numbers:**
   $$
   x^2 + y^2 = 100
   $$

3. **Sum of the numbers:**
   We need to verify that the sum $ x + y = 14 $.

To find the sum of the numbers, we can use the identity:
$$
(x + y)^2 = x^2 + y^2 + 2xy
$$

Substituting the known values:
$$
(x + y)^2 = 100 + 2 \times 48
$$
$$
(x + y)^2 = 100 + 96
$$
$$
(x + y)^2 = 196
$$

Taking the square root of both sides:
$$
x + y = \sqrt{196} = 14
$$

Thus, the sum of the two numbers is indeed 14, which corresponds to Option 1.

**Explanation for why other options are incorrect:**

- **Option 2 (16):** If the sum were 16, then $(x + y)^2$ would be 256, which does not match the calculated value of 196.
  
- **Option 3 (19):** If the sum were 19, then $(x + y)^2$ would be 361, which is again incorrect compared to 196.

- **Option 4 (24):** If the sum were 24, then $(x + y)^2$ would be 576, which is far from the correct value of 196.

Therefore, based on the calculations and logical reasoning, Option 1 is the correct answer, as it satisfies all the given conditions of the problem.