**Given Information:**

- The area of two similar triangles: 49 cm² and 64 cm²
- The difference of the corresponding altitudes: 10 cm

**Let's Solve the Problem Step-by-Step:**

**Step 1: Understand the Concept of Similar Triangles**

Similar triangles are two triangles that have the same shape but possibly different sizes. The corresponding angles of similar triangles are equal, and the corresponding sides are proportional.

**Step 2: Understand the Relationship between Areas of Similar Triangles**

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. This can be represented as:

(area of triangle 1) / (area of triangle 2) = (side 1 of triangle 1 / side 1 of triangle 2)²

**Step 3: Set up the Ratio of Areas**

Let's set up the ratio of the areas of the given triangles:

49 cm² / 64 cm² = (side 1 of triangle 1 / side 1 of triangle 2)²

**Step 4: Solve for the Ratio of Sides**

To find the ratio of the sides, we take the square root of the ratio of the areas:

√(49/64) = side 1 of triangle 1 / side 1 of triangle 2

**Step 5: Find the Ratio of the Altitudes**

Since the altitudes of two similar triangles are proportional to their corresponding sides, we can say:

(side 1 of triangle 1 / side 1 of triangle 2) = (altitude 1 of triangle 1 / altitude 1 of triangle 2)

**Step 6: Set up the Equation**

Let's set up the equation using the given difference of the corresponding altitudes:

(side 1 of triangle 1 / side 1 of triangle 2) = (altitude 1 of triangle 1 - altitude 1 of triangle 2) / altitude 1 of triangle 2

**Step 7: Substitute the Values**

Substituting the values we found earlier:

√(49/64) = (altitude 1 of triangle 1 - altitude 1 of triangle 2) / altitude 1 of triangle 2

**Step 8: Solve for the Altitudes**

Let's solve the equation for the altitudes:

√(49/64) = (altitude 1 of triangle 1 - altitude 1 of triangle 2) / altitude 1 of triangle 2

√(49/64) * altitude 1 of triangle 2 = altitude 1 of triangle 1 - altitude 1 of triangle 2

√(49/64) * altitude 1 of triangle 2 + altitude 1 of triangle 2 = altitude 1 of triangle 1

Simplifying further:

(√49 * altitude 1 of triangle 2 + 8 * altitude 1 of triangle 2) / 8 = altitude 1 of triangle 1

(7 * altitude 1 of triangle 2 + 8 * altitude 1 of triangle 2) / 8 = altitude 1 of triangle 1

15 * altitude 1 of triangle 2 / 8 = altitude 1 of triangle 1

15 * altitude 1 of triangle 2 = 8 * altitude 1 of triangle