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Advanced Math / Nonlinear functions Difficulty: Hard

A submersible device is used for ocean research. The function $g (x) = - \frac{1}{55} (x + 19) (x - 35)$ gives the depth below the surface of the ocean, in meters, of the submersible device $x$ minutes after collecting a sample, where $x greater than 0$ . How many minutes after collecting the sample did it take for the submersible device to reach the surface of the ocean?

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Explanation

The correct answer is $35$ . It's given that the function $g (x) = - \frac{1}{55} (x + 19) (x - 35)$ gives the depth below the surface of the ocean, in meters, of the submersible device $x$ minutes after collecting a sample, where $x greater than 0$ . It follows that when the submersible device is at the surface of the ocean, the value of $g (x)$ is $0$ . Substituting $0$ for $g (x)$ in the equation $g (x) = - \frac{1}{55} (x + 19) (x - 35)$ yields $0 = - \frac{1}{55} (x + 19) (x - 35)$ . Multiplying both sides of this equation by $negative 55$ yields $0 = (x + 19) (x - 35)$ . Since a product of two factors is equal to $0$ if and only if at least one of the factors is $0$ , either $x + 19 = 0$ or $x - 35 = 0$ . Subtracting $19$ from both sides of the equation $x + 19 = 0$ yields $x = - 19$ . Adding $35$ to both sides of the equation $x - 35 = 0$ yields $x equals 35$ . Since $x greater than 0$ , $35$ minutes after collecting the sample the submersible device reached the surface of the ocean.