# Question #5ebcb

Feb 4, 2018

The equation is $120 = 2 v + 10$

The original volume was $55 \textcolor{w h i t e}{\text{-""dB}}$

#### Explanation:

First of all, the decibel scale is NOT linear like this problem assumes, but that's a whole different discussion. For now, just assume that doubling the original volume also multiplies the dB measurement by 2.

In that case, then this is how to write the word problem as an algebraic problem and solve for the original volume.

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Let $v$ be the original volume.

${\overbrace{\text{The new volume would be 120")^(120)overbrace(" if you }}}^{=}$

${\overbrace{\text{ double the original volume ")^(2 xx v) overbrace(" and add 10dB}}}^{+ 10}$

So putting that all together as an equation, we get:

$120 = 2 \times v + 10$

$120 = 2 v + 10$

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Now to solve the equation:

$120 = 2 v + 10$

Subtract $\textcolor{red}{10}$ from both sides.

$120 - \textcolor{red}{10} = 2 v + \cancel{10} - \cancel{\textcolor{red}{10}}$

$110 = 2 v$

Divide both sides by $\textcolor{b l u e}{2}$.

$\frac{110}{\textcolor{b l u e}{2}} = \frac{\cancel{2} v}{\cancel{\textcolor{b l u e}{2}}}$

$55 = v$

So the original volume is $55 \textcolor{w h i t e}{\text{-""dB}}$.