A research ship lowers a probe into the ocean to explore an underwater trench. The probe starts on a crane 16 feet above the surface and is lowered at a rate of 9.7 feet per minute. Write an expression for the depth of the probe after m minutes. What is the depth of the probe after 60 ​minutes? Write an expression for the depth of the probe after m minutes.

Question

depth =

Answer 1

To find an expression for the depth of the probe after \( m \) minutes, we start with the initial position of the probe.

1. The probe starts at a height of 16 feet above the surface.
2. It is being lowered into the ocean at a rate of 9.7 feet per minute.

The depth \( d \) of the probe after \( m \) minutes can be calculated using the formula:

\[ d = 16 + 9.7m \]

However, since the probe is being lowered, the depth will actually increase as it descends into the water, meaning we should consider the depth as a negative change since we start above the water and lower into a negative depth.

Thus, the expression for the depth after \( m \) minutes should be written as:

\[ d = 9.7m - 16 \]

Now, let's find the depth of the probe after \( 60 \) minutes:

\[ d = 9.7(60) - 16 \]

Calculating this step-by-step:

1. Calculate \( 9.7 \times 60 \): \[ 9.7 \times 60 = 582 \]

2. Then subtract 16: \[ d = 582 - 16 = 566 \]

Therefore, the depth of the probe after \( 60 \) minutes is:

\[ \text{Depth} = 566 \text{ feet} \]

Answer 2

wrong

Answer 3

I apologize for the misunderstanding in my previous response. Let's clarify:

1. The probe starts 16 feet above the water's surface.
2. It is being lowered into the water at a rate of 9.7 feet per minute.

This means that as time passes, the probe is moving deeper into the ocean, and the depth will increase negatively relative to the water surface.

To correctly express the depth \( d \) of the probe after \( m \) minutes, we should take into account that we are adding the depth that the probe has descended, but since it starts above the surface, we should subtract the height above the surface:

\[ d = 9.7m - 16 \]

However, to represent the depth (which is considered a positive quantity downwards) correctly, we should adjust the equation:

The correct formula for the depth \( d \) (as a positive quantity) after \( m \) minutes is:

\[ d = 16 + 9.7m \]

But, since it's going down into the ocean, we need the expression to represent the total depth below the surface, therefore:

1. After \( m \) minutes, the probe is lowered:
   • The amount it descends is \( 9.7m \)
2. The probe started at 16 feet above the water surface, and we want the depth to be positive below the surface, leading to the adjustment:

The total depth as a function of time should be:

\[ d = 16 + 9.7m \]

Now, to find the depth after 60 minutes:

Substituting \( m = 60 \):

\[ d = 16 + 9.7 \times 60 \]

Calculating \( 9.7 \times 60 \):

\[ 9.7 \times 60 = 582 \]

Now, adding that to 16:

\[ d = 16 + 582 = 598 \]

Therefore, the correct depth of the probe after 60 minutes is:

\[ \text{Depth} = 598 \text{ feet} \]

Thank you for your patience, and I hope this clears up the confusion!

Answer 4

no

Answer 5

I apologize for the confusion earlier. Let's carefully analyze the problem and provide the correct expression for the depth of the probe.

Given:

• The probe starts 16 feet above the surface.
• It is lowered at a rate of 9.7 feet per minute.

Understanding the Problem:

• Since the probe is above the water's surface initially (16 feet above), as it descends, the total depth below the surface will be calculated based on how far it has descended.

Expression for Depth:

To write the expression for the depth \( d \) of the probe after \( m \) minutes:

1. After \( m \) minutes, the probe has descended \( 9.7m \) feet.
2. Initially, it was 16 feet above the surface, which means we need to account for this by effectively subtracting this height from the total depth below the surface.

Thus, the correct expression for the depth \( d \) after \( m \) minutes is:

\[ d = 9.7m - 16 \]

Calculate the Depth after 60 Minutes:

Now, substituting \( m = 60 \):

\[ d = 9.7(60) - 16 \] Calculating \( 9.7 \times 60 \):

\[ 9.7 \times 60 = 582 \]

Then, substituting back:

\[ d = 582 - 16 \] \[ d = 566 \]

Conclusion:

The depth of the probe after 60 minutes is:

\[ \text{Depth} = 566 \text{ feet} \]

I appreciate your patience, and I hope this clarifies the calculations correctly! Thank you for allowing me to correct the explanation.