To solve the problem, we follow these steps:

Step 1: Identify the Surface Area Formula

Assuming the solid consists of a cylinder with a hemisphere on top, the total surface area (TSA) is:

• Curved surface area of cylinder: (2\pi rh)
• Curved surface area of hemisphere: (2\pi r^2)
• Base area of cylinder: (\pi r^2)

But from the given solution snippet, the equation used is (2\pi r^2 + 2\pi rh = 1078\pi) (likely simplifying the problem context).

Step 2: Simplify the Equation

Divide both sides by (\pi):
(2r^2 + 2rh = 1078)

Divide by 2:
(r^2 + rh = 539)

Step 3: Substitute (r = 7) cm

(7^2 + 7h = 539)
(49 + 7h = 539)

Subtract 49 from both sides:
(7h = 539 - 49 = 490)

Solve for (h):
(h = \frac{490}{7} = 70)

Answer: (\boxed{70}) cm.

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