Take log: t × log(0.95) < log(0.25) → t > log(0.25)/log(0.95) ≈ (−1.3863)/(−0.05129) ≈ <<1.3863/0.05129≈27.03>>27.03
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- after 2 hr: 19 × 0.95 = 18.05
- This is geometric: E = 20 × (0.95)^t
- Solve: 20 × (0.95)^t < 5 → (0.95)^t < 0.25
- So t = 28 full hours.
- #### 28
- A science policy analyst evaluates a city’s carbon capture system that removes 12 tons of CO₂ in the first month, increasing by 8% each month due to machine optimization. How many tons are removed in total over the first 6 months, to the nearest ton?
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\(\boxed{1}\)**Question:
A quantum biologist is modeling the interaction between quantum states and genetic mutations. Given the expression $x^4 - 5x^2 + 4$, factor it completely.
The expression $x^4 - 5x^2 + 4$ is a quadratic in terms of $x^2$. Let $u = x^2$. The expression becomes $u^2 - 5u + 4$. We need to factor this quadratic.
Find two numbers that multiply to $4$ (the constant term) and add to $-5$ (the coefficient of the linear term).
The numbers $-4$ and $-1$ satisfy these conditions since $(-4) \times (-1) = 4$ and $(-4) + (-1) = -5$.
Thus, $u^2 - 5u + 4$ factors as $(u - 4)(u - 1)$.
Substitute back $u = x^2$:
Both $x^2 - 4$ and $x^2 - 1$ are differences of squares:
x^2 - 4 = (x - 2)(x + 2)
Therefore, the complete factorization is: