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Home » After 4 hours: \( 500 \times 2^4 = 8000 \) - High Altitude Science

After 4 hours: \( 500 \times 2^4 = 8000 \) - High Altitude Science

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 L(m) = m^2 - 2m \cdot m + m^2 + 4 = 2m^2 - 2m^2 + 4 = 4

📌 So the minimum value is always 4, independent of \( m \). But the problem asks for the minimum loss to be 1, which is impossible under this function form.

📌 Wait — re-examining: perhaps a typo in interpretation. Let’s suppose the function is \( L(w) = (w - m)^2 + 1 \), so minimum is 1. But the given is \( L(w) = w^2 - 2mw + m^2 + 4 = (w - m)^2 + 4 - m^2 + m^2 = (w - m)^2 + 4 \)? No:

📌 Thus minimum value is \( 0 + 4 = 4 \), always. So minimum cannot be 1.

📌 But the problem says "find \( m \) such that the minimum loss is exactly 1". This is impossible unless the constant term is adjusted.

📌 Wait — perhaps the function is \( L(w) = w^2 - 2mw + m^2 + c \), and \( c = 1 \), and we find minimum. But in the problem, it's written as \( +4 \). Let’s assume it's correct and recompute.

📌 Given \( L(w) = w^2 - 2mw + m^2 + 4 = (w - m)^2 + 4 \), minimum is always 4. So no \( m \) makes it 1.

📌 But perhaps the problem meant: find \( m \) such that the **minimum value** is 1, which implies a different constant. Let’s suppose there is a typo and the constant is variable: \( L(w) = w^2 - 2mw + m^2 + c \). Then minimum is \( c \), and we set \( c = 1 \). But the problem fixes \( +4 \), so \( c = 4 \).

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