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Home » \( x^2 + (x + 6)^2 = 15^2 \) - Sterling Industries

\( x^2 + (x + 6)^2 = 15^2 \) - Sterling Industries

📅 April 15, 2026 👤 scraface

Apr 15, 2026

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📌 A geologist has 3 distinct types of minerals: quartz, feldspar, and mica. If she randomly picks 3 minerals from a collection containing 4 quartz, 5 feldspar, and 6 mica, what is the probability that she picks exactly 1 mineral of each type?

📌 Next, calculate the number of favorable outcomes where exactly 1 mineral of each type is chosen. This involves choosing 1 quartz from 4, 1 feldspar from 5, and 1 mica from 6:

📌 The probability of picking exactly one of each type of mineral is the ratio of favorable outcomes to the total number of outcomes:

📌 An entomologist is studying a population of insects consisting of 7 beetles, 5 ants, and 8 butterflies. If she randomly selects 4 insects, what is the probability that she selects at least one of each type?

📌 Next, calculate the number of ways to choose 4 insects such that at least one of each type is included. We can break this down into cases:

📌 Add these cases together to find the total number of favorable outcomes:

📌 The probability that she selects at least one of each type is:

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$$ 4 = -a(0 - 3)^2 + 10 \implies 4 = 9a + 10 \implies 9a = -6 \implies...
However, since the coefficient of $ x^2 $ is given as $ -a $, and para...
$$ 4 = -a(9) + 10 \implies -9a = 6 \implies a = -rac{2}{3} \quad ext...
Re-evaluating, the standard form $ y = -ax^2 + bx + c $ with vertex $...
$$ 10 = -9a + 18a + 4 \implies 10 = 9a + 4 \implies 9a = 6 \implies a...
Thus, the value of $ a $ is $ oxed{\dfrac{2}{3}} $.
Question: An entomologist studies the growth of a bee colony, modeled...
Solution: Factor the numerator:
$$ rac{t^2 + 5t + 6}{t + 2} = rac{(t + 2)(t + 3)}{t + 2} = t + 3 \qu...
eq -2 ext{)}. $$

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