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The problem in the image states:

"If the product of three positive continued proportional numbers is 343, then what is the median proportional of the first and third?"

Let the three continued proportional numbers be \( a \), \( b \), and \( c \). In continued proportion, the second number \( b \) is the geometric mean of the first \( a \) and third \( c \), meaning:
\[
b^2 = a \cdot c
\]

Additionally, the product of these three numbers is given as:
\[
a \cdot b \cdot c = 343
\]

Using the relation \( b^2 = a \cdot c \), we can express \( a \cdot c \) as \( b^2 \). Substituting this into the product equation gives:
\[
b \cdot b^2 = 343
\]
\[
b^3 = 343
\]

Taking the cube root of both sides:
\[
b = \sqrt[3]{343} = 7
\]

Thus, the median proportional (the second number \( b \)) is 7.

The correct answer is:
**(A) 7**

Would you like more details or have any questions?

Here are five related questions:
1. How do you find the geometric mean of two numbers?
2. What are the properties of continued proportion?
3. How can you solve other problems involving continued proportions?
4. What is the cube root of a number, and how is it calculated?
5. How would the solution change if the product of the numbers was different?

**Tip:** For problems involving proportions, recognizing relationships like geometric means can simplify the solution process greatly.

If the product of three positive continued proportional numbers is 343, then what is the median proportional of the first and third?

Solve a problem involving continued proportions where the product of three positive numbers is 343. Learn how to find the median proportional using the geometric mean and cube root. Suitable for high school students.

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