Question

9 mark for review 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 $x > 0$. how many minutes after collecting the sample did it take for the submersible device to reach the surface of the ocean?

Explanation:

Step1: Understand the problem

The function \( g(x) = -\frac{1}{55}(x + 19)(x - 35) \) represents the depth of the submersible device below the ocean surface (in meters) \( x \) minutes after collecting a sample. We need to find the time \( x \) when the device reaches the surface, which means the depth \( g(x) = 0 \) (since at the surface, the depth is 0 meters). Also, we know that \( x>0 \).

Step2: Set up the equation

Set \( g(x) = 0 \):

$$ -\frac{1}{55}(x + 19)(x - 35)=0 $$

Since \( -\frac{1}{55}
eq0 \), we can focus on the product \( (x + 19)(x - 35)=0 \) (because if a product of factors is zero, at least one of the factors must be zero).

Step3: Solve for \( x \)

Using the zero - product property:

• If \( x + 19 = 0 \), then \( x=-19 \). But we are given that \( x > 0 \), so this solution is not valid.
• If \( x - 35=0 \), then \( x = 35 \). This value of \( x \) satisfies the condition \( x>0 \).

Answer:

35