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Charles P.
Online MS Math Instructor / Tutor 9+ years
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Pre-Calculus
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Question:

Where does the half life formula come from?\\ \ \*

Charles P.

\centering \underline{half-life}\\ \vspace{-.5cm} \begin{align*} A(t) &= A_0e^{-kt}\\ \frac{1}{2} \bcancel{A_0} &= \bcancel{A_0}e^{-kt}\\ \ln\left(2^{-1}\right) &= \ln\left(e^{-kt}\right)\\ -\ln2 &= -kt\\ h &= t =\fbox{$\dfrac{\ln 2}{k}$}\\ \end{align*}

Calculus
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Question:

A pebble is dropped from a hot air balloon. Find how far it is falling, and its acceleration after 3.5 sec. let $s(t) = 16t^2$, where $t$ is in seconds and $s$ is in feet.\\ \ \*

Charles P.

\begin{align*} s(3.5) &= 16(3.5)^2 = \fbox{196 \text{ ft }}\\ v(3.5) &= s'(3.5) = 32(3.5) = \fbox{112 ft/s}\\ a(3.5) &= v'(3.5) = s''(3.5) = \fbox{32 ft/$s^2$}\\ \end{align*}

Algebra
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Question:

A baseball is hit at a point $3$ feet above the ground at a velocity of $100$ feet per second and at an angle of $45^o$ with respect to the ground. The path of the ball is given by the function $f(x) = -0.0032x2 + x + 3$, where $f(x)$ is the height of the baseball (in feet) and x is the horizontal distance from home plate (in feet). What is the maximum height reached by the baseball?

Charles P.

$$h=\dfrac{-b}{2a} = \dfrac{-1}{2(-0.0032)} = 156.25$$, $$k = \text{ height } = \fbox{81.125 feet}$$

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