Questions: The function, f(x), describes the height of a dome on top of a building, where f(x) is the height from the base of the dome and x is the horizontal distance from where the dome meets the building. The domain of the function is ≤ x ≤

Solution Steps

The function given is \( f(x) = 2\sqrt{-x^2 + 10x} \). This function describes the height of a dome on top of a building, where \( f(x) \) is the height from the base of the dome and \( x \) is the horizontal distance from where the dome meets the building.

To find the domain of the function, we need to ensure that the expression inside the square root is non-negative because the square root of a negative number is not defined in the real number system.

Set the expression inside the square root to be greater than or equal to zero: \[ -x^2 + 10x \geq 0 \]

Solve the inequality \( -x^2 + 10x \geq 0 \):

1. Factor the quadratic expression: \[ -x^2 + 10x = x(10 - x) \]

2. Set the factored expression greater than or equal to zero: \[ x(10 - x) \geq 0 \]

3. Find the critical points by setting each factor to zero: \[ x = 0 \] \[ 10 - x = 0 \implies x = 10 \]

4. Determine the intervals to test: \[ x \in [0, 10] \]

Final Answer