Question

solve. An oil drilling crew estimated the depth of an oil pocket to be \(-450\) feet. After drilling to three times that depth, they finally found oil. At what depth did they find oil?

Short answer

Answer: The oil drilling crew found oil at a depth of -1350 feet.

Step-by-step solution

Determine the estimated depth of the oil pocket

The oil pocket is estimated to be at a depth of \(-450\) feet.

Calculate the depth to which they drilled down

The drilling crew drilled down to three times the estimated depth. So, we need to multiply \(-450\) feet by 3: \(-450 \times 3 = -1350\) feet.

Determine the depth at which they found oil

The depth at which they found oil is the same as the depth to which they drilled down, which is \(-1350\) feet. So, the oil drilling crew found oil at a depth of \(-1350\) feet.

Key concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign (e.g., -3, -450). In mathematics, negative numbers are used to denote a decrease, or indicate a value below a certain reference point. For example:

• An altitude of -450 feet means 450 feet below sea level.
• A temperature of -10 °C represents 10 degrees below the freezing point of water.
• In financial contexts, a negative balance suggests debt or money owed.

Working with negative numbers requires understanding signed values. Pairing negative numbers with positive numbers involves careful arithmetic. When adding a negative and a positive number, consider the signs to determine if the numbers subtract or add. Similarly, subtracting a negative number actually means adding its positive counterpart.

Multiplication of Integers

When multiplying integers, the rules for negative and positive signs affect the product’s sign. The rules are:

• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative
• Negative × Positive = Negative

Thus, multiplying -450 by 3 becomes:

• First, identify the numbers: -450 and 3.
• One is negative, and the other is positive.
• The result is negative, as the rules indicate Negative × Positive = Negative.

The multiplication yields -1350, emphasizing the downward direction in depth calculation.

Depth Calculation in Real-World Contexts

Calculating depth involves understanding direction and given quantities, especially when negative numbers play a role. Depth is often expressed as a negative value to indicate below a reference point such as sea level. For example, if an oil drilling crew estimates an oil pocket at -450 feet, they indicate it is 450 feet below sea level. In the given problem, the drilling process reached a depth of -1350 feet, found by calculating how much further down they needed to go. This entails multiplying the initially estimated depth ( -450 feet) by three, reflecting they drilled three times deeper to reach the oil pocket. Such calculations are pivotal in engineering and geology to prevent errors and ensure effective operations.