$256^b t^2=4^2-{ }^{2 b}

Rewrite the given equation in terms of powers of 4.
Substitute x = 4 2 b to simplify the equation.
Solve for t 2 in terms of x and find the condition for t 2 to be non-negative, leading to b < 1 .
Express t in terms of b : t = ± 4 4 b 16 − 4 2 b ​ ​ .
The relationship between b and t is t = ± 4 4 b 16 − 4 2 b ​ ​ ​ .

Explanation

Problem Analysis We are given the equation 25 6 b t 2 = 4 2 − 4 2 b and we want to find the relationship between b and t .

Rewriting the Equation First, rewrite the equation using powers of 4. Since 256 = 4 4 , we have ( 4 4 ) b t 2 = 4 2 − 4 2 b , which simplifies to 4 4 b t 2 = 16 − 4 2 b .

Substitution Let x = 4 2 b . Then the equation becomes x 2 t 2 = 16 − x .

Solving for t^2 Now, solve for t 2 in terms of x : t 2 = x 2 16 − x ​ .

Non-negativity Condition Since t 2 must be non-negative, we have x 2 16 − x ​ ≥ 0 . Since 0"> x 2 > 0 , we must have 16 − x ≥ 0 , which means x ≤ 16 .

Finding the bound for b Substitute back x = 4 2 b to get 4 2 b ≤ 16 , which means 4 2 b ≤ 4 2 . Thus, 2 b ≤ 2 , so b ≤ 1 .

Refining the bound for b Also, since 0"> x = 4 2 b > 0 , we must have 0"> 16 − x > 0 for t to be real, so x < 16 . Thus 4 2 b < 16 , so 2 b < 2 , and b < 1 .

Expressing t in terms of b Express t in terms of b : t = ± 4 4 b 16 − 4 2 b ​ ​ = ± ( 4 2 b ) 2 16 − 4 2 b ​ ​ .

Final Answer Therefore, the relationship between b and t is t = ± 4 4 b 16 − 4 2 b ​ ​ .


Examples
This relationship can be used in physics to describe damped oscillations, where 'b' might represent a damping coefficient and 't' represents time. The equation shows how the amplitude of the oscillation changes over time depending on the damping. For example, in a spring-mass system with damping, the position of the mass can be modeled using such equations, where 'b' affects how quickly the oscillations die down, and 't' determines the position at a given time. Understanding this relationship helps engineers design systems with controlled damping, such as shock absorbers in cars or vibration isolators in machinery.

Answered by GinnyAnswer | 2025-07-03

The equation can be rewritten in terms of powers of 4, leading to a simplified expression for t 2 . The conditions for t 2 to be non-negative imply that b must be less than 1. The final relationship shows t as a function of b .
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Answered by Anonymous | 2025-07-04