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Home » Substance B : 8 moles / 2 = 4 lots - Get link 4share

Substance B : 8 moles / 2 = 4 lots - Get link 4share

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Question: For all real numbers \( a, b, c \), find the number of functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( f(a + b) = f(a) + f(b) \) and \( f \) is linear.

📌 Solution: A function \( f: \mathbb{R} \to \mathbb{R} \) is linear if it can be expressed as \( f(x) = kx \) for some constant \( k \). Given the functional equation:

📌 \[ f(a + b) = f(a) + f(b), \]

📌 assuming linearity, the function satisfies this by the very definition of linearity (additivity and homogeneity). Thus, the only functions meeting the criteria are those of the form \( f(x) = kx \). Since \( k \) can be any real number, there are infinitely many such linear functions, one for each real number \( k \). Therefore, the number of such functions is:

📌 \boxed{\infty}

📌 Question: The perimeter of a rectangle is 50. What is the largest possible area of the rectangle?

📌 Solution: Let the length and width of the rectangle be \( l \) and \( w \), respectively. The perimeter is given by:

📌 \[ 2l + 2w = 50 \quad \Rightarrow \quad l + w = 25. \]

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This is a binomial probability problem where we have \( n = 7 \) trial...
The formula for the binomial probability is:
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
Substitute \( n = 7 \), \( k = 4 \), and \( p = \frac{1}{2} \):
P(X = 4) = \binom{7}{4} \left( \frac{1}{2} \right)^4 \left( \frac{1}{2...
Calculate \( \binom{7}{4} \):
\binom{7}{4} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
So, the probability becomes:
P(X = 4) = 35 \times \left( \frac{1}{2} \right)^7 = 35 \times \frac{1}...
Thus, the probability that exactly 4 stations detect an earthquake is...

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