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Chapter 3: Problem 44

Suppose the effects of detonating a nuclear bomb will be felt over a distance from the point of detonation that is directly proportional to the cube root of the yield of the bomb. Suppose a 100 -kiloton bomb has certain effects to a radius of \(3 \mathrm{km}\) from the point of detonation. Find the distance to the nearest tenth that the effects would be felt for a 1500 -kiloton bomb.

Short Answer

Expert verified
The distance is approximately 7.4 km.

Step by step solution

01

- Understand the Proportionality Relationship

The distance over which the effects are felt is directly proportional to the cube root of the bomb's yield. This can be written mathematically as: \[ d = k \times \root{3}\big{(}Y\big{)} \] where \(d\) is the distance, \(Y\) is the yield, and \(k\) is the proportionality constant.
02

- Determine the Proportionality Constant

Given that a 100-kiloton bomb affects a radius of 3 km, substitute these values into the equation to solve for \(k\): \[ 3 = k \times \root{3}\big{(}100\big{)} \] The cube root of 100 is approximately 4.64: \[ 3 = k \times 4.64 \] Solve for \(k\): \[ k = \frac{3}{4.64} \ k \approx 0.65 \]
03

- Apply the Proportionality Constant to the 1500-kiloton Bomb

Now use the constant \(k\) to find the distance for a 1500-kiloton bomb: \[ d = 0.65 \times \root{3}\big{(}1500\big{)} \] The cube root of 1500 is approximately 11.45: \[ d = 0.65 \times 11.45 \ d \approx 7.44 \text{km} \]
04

- Round to the Nearest Tenth

Finally, round the distance to the nearest tenth: \[ 7.44 \text{km} \approx 7.4 \text{km} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cube Root
The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For instance, the cube root of 8 is 2, because 2 \( \times \) 2 \( \times \) 2 = 8. In mathematical notation, the cube root of a number \( Y \) is expressed as \( \root{3}\big{(}Y\big{)} \). Finding the cube root is essential in problems involving volumes and certain proportionality relationships, as seen in the given exercise where the distance is directly proportional to the cube root of the bomb's yield.
Yield Calculation
Yield indicates the explosive power of a bomb, measured in kilotons. Kilotons quantify the equivalent amount of TNT required to produce a similar explosion. In our exercise, you calculate the impact distance based on the yield. Calculating yield involves taking measurable factors like energy release or pressure impact from the detonation. Determining exact output is complex, which is why simpler proportional relationships like cube root are helpful in estimations and predictions, particularly for practical applications in physics and engineering.
Proportionality Constant
The proportionality constant \( k \) links two directly proportional quantities. For the exercise, it connects the distance \( d \) with the cube root of the bomb's yield \( \root{3}\big{(}Y\big{)} \). To determine \( k \), you initially use known values. Given that a 100-kiloton bomb impacts within 3 km, you substitute these into the formula, solving: \[ 3 = k \times \root{3}\big{(}100\big{)} \ \] Knowing the cube root of 100 is approximately 4.64, you then solve: \[ k = \frac{3}{4.64} \ \] This yields \( k \approx 0.65 \). This constant helps predict impacts for different yields, ensuring consistency in your calculations.
Distance Calculation
Distance calculation involves using the previously found proportionality constant to determine how far the effects of an explosion will spread. For a 1500-kiloton bomb, you apply: \[ d = k \times \root{3}\big{(}1500\big{)} \ \] With \( k \approx 0.65 \) and the cube root of 1500 being 11.45: \[ d = 0.65 \times 11.45 \approx 7.44 \text{km} \ \] Finally, you round this to the nearest tenth (7.4 km). This method provides a systematic way to estimate distances based on explosive yields, important in planning safety measures and resource deployment in disaster preparedness.

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