Solve each problem. Nuclear Bomb DetonationSuppose 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 km from the point of detonation. Find, to the nearest tenth, the dis-tance over which the effects would be felt for a 1500-kiloton bomb.

Verified step by step guidance

1. Identify the relationship: The distance over which the effects are felt is directly proportional to the cube root of the yield of the bomb.
2. Express the relationship mathematically: Let d be the distance and y be the yield. Then, d = k \cdot y^{1/3}, where k is the proportionality constant.
3. Use the given information to find k: For a 100-kiloton bomb, d = 3 km. So, 3 = k \cdot 100^{1/3}.
4. Solve for k: k = \frac{3}{100^{1/3}}.
5. Use the value of k to find the distance for a 1500-kiloton bomb: d = k \cdot 1500^{1/3}.

Key Concepts

Proportional Relationships

A proportional relationship indicates that two quantities maintain a constant ratio. In this context, the distance affected by the bomb's detonation is directly proportional to the cube root of its yield. This means that if one quantity increases, the other increases in a predictable manner, allowing us to set up a ratio to solve for unknown values.

Cube Root Function

The cube root function is a mathematical operation that finds a number which, when multiplied by itself three times, gives the original number. In this problem, the distance affected by the bomb is determined by taking the cube root of the bomb's yield. Understanding how to compute and interpret cube roots is essential for solving the problem accurately.

Unit Conversion and Scaling

Unit conversion and scaling involve adjusting measurements to ensure consistency in units. In this scenario, we need to relate the yield of the bomb in kilotons to the distance in kilometers. Properly scaling the values based on the proportional relationship allows us to calculate the new distance affected by the larger bomb yield accurately.