Nuclear Bomb Detonation 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 km from the point of detonation. Find, to the nearest tenth, the distance over which the effects would be felt for a 1500-kiloton bomb.

Verified step by step guidance

1. Identify the relationship given: the distance over which effects are felt, \(d\), is directly proportional to the cube root of the yield, \(y\). This can be written as \(d = k \cdot y^{\frac{1}{3}}\), where \(k\) is the constant of proportionality.
2. Use the information about the 100-kiloton bomb to find \(k\). Substitute \(d = 3\) km and \(y = 100\) kilotons into the equation: \$3 = k \cdot 100^{\frac{1}{3}}$.
3. Solve for \(k\) by isolating it: \(k = \frac{3}{100^{\frac{1}{3}}}\).
4. Now, use the value of \(k\) to find the distance for the 1500-kiloton bomb. Substitute \(y = 1500\) into the original formula: \(d = k \cdot 1500^{\frac{1}{3}}\).
5. Calculate the cube roots and multiply by \(k\) to find \(d\), then round the result to the nearest tenth to get the distance over which the effects would be felt.

Key Concepts

Direct Proportionality

Direct proportionality means one quantity changes at a constant rate relative to another. If the distance affected is directly proportional to the cube root of the bomb's yield, then distance = k × (yield)^(1/3), where k is a constant. Understanding this relationship helps set up the equation to find unknown distances.

Cube Root Function

The cube root function, denoted as ∛x, is the inverse of cubing a number. It extracts the value that, when cubed, returns the original number. In this problem, the distance depends on the cube root of the bomb's yield, so calculating cube roots is essential to relate yield to distance.

Solving Proportional Relationships Using Given Data

To find the constant of proportionality, use known values (100-kiloton bomb and 3 km radius). Then apply this constant to the new yield (1500 kilotons) to find the unknown distance. This process involves substituting values, solving for the constant, and then calculating the desired quantity.