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Home » This is a sphere centered at $(1, 0, 0)$ with radius 1. - Tacotoon

This is a sphere centered at $(1, 0, 0)$ with radius 1. - Tacotoon

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 An angel investor is evaluating the growth potential of a tech startup. The companyâs revenue \( R(t) \) over time \( t \) (in years) is modeled by the function \( R(t) = 3t^3 - 5t^2 + 2t + 10 \). Determine the time \( t \) when the revenue growth rate is maximized.

📌 To find the time when the revenue growth rate is maximized, we first determine the derivative of the revenue function \( R(t) \) to find the growth rate:

📌 R'(t) = rac{d}{dt}(3t^3 - 5t^2 + 2t + 10) = 9t^2 - 10t + 2.

📌 Next, we find the critical points by setting \( R'(t) = 0 \):

📌 9t^2 - 10t + 2 = 0.

📌 Using the quadratic formula \( t = rac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 9 \), \( b = -10 \), and \( c = 2 \), we calculate:

📌 t = rac{10 \pm \sqrt{(-10)^2 - 4 \cdot 9 \cdot 2}}{2 \cdot 9} = rac{10 \pm \sqrt{100 - 72}}{18} = rac{10 \pm \sqrt{28}}{18}.

📌 Simplifying further:

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$ b_3 = M(b_2) = \frac{2}{3} - \frac{(\frac{2}{3})^3}{3} = \frac{2}{3}...
$ b_4 = M(b_3) = \frac{46}{81} - \frac{(\frac{46}{81})^3}{3} $
First, compute $ \left( \frac{46}{81} \right)^3 = \frac{97336}{531441}...
b_4 = \frac{46}{81} - \frac{1}{3} \cdot \frac{97336}{531441} = \frac{4...
Convert $ \frac{46}{81} $ to denominator $ 1594323 $:
\frac{46}{81} = \frac{46 \cdot 19683}{1594323} = \frac{904518}{1594323...
b_4 = \frac{904518 - 97336}{1594323} = \frac{807182}{1594323}
Thus, $ b_4 = \boxed{\dfrac{807182}{1594323}} $.
Question: A science policy analyst studying energy output models a sys...
Solution: Let $ u = \sqrt{x} $, so $ x = u^2 $, and $ P(x) = u^2 \cdot...

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