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We are now in a position to study some applications of quadratic equations. Quadratic equations can arise from a variety of physical (applied) and mathematical (logical) problems.
We will, again, apply the five-step method for solving word problems.
Remember, step 1 is very important.
ALWAYS START BY INTRODUCING A VARIABLE.
Once the quadratic equation is developed (step 2), try to solve it by factoring. If factoring doesn’t work, use the quadratic formula. A calculator may help to make some of the calculations a little less tedious.
A producer of personal computer mouse covers determines that the number $N$ of covers sold is related to the price $x$ of a cover by $N=35x-{x}^{2}.$ At what price should the producer price a mouse cover in order to sell 216 of them?
$$\begin{array}{l}\begin{array}{ccc}\text{Step\hspace{0.17em}}1:& & \text{Let}\text{\hspace{0.17em}}x=\text{the}\text{\hspace{0.17em}}\text{price}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{mouse}\text{\hspace{0.17em}}\text{cover}\text{.}\end{array}\hfill \\ \begin{array}{lll}\text{Step\hspace{0.17em}}2:\hfill & \hfill & \text{Since}\text{\hspace{0.17em}}N\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}\text{be}\text{\hspace{0.17em}}\text{216,}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{equation}\text{\hspace{0.17em}}\text{is}\hfill \\ \hfill & \hfill & 216=35x-{x}^{2}\hfill \end{array}\hfill \\ \begin{array}{lllllll}\text{Step\hspace{0.17em}}3:\hfill & \hfill & 216\hfill & =\hfill & 35x-{x}^{2}\hfill & \hfill & \text{Rewrite}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{standard}\text{\hspace{0.17em}}\text{form}\text{.}\hfill \\ \hfill & \hfill & {x}^{2}-35x+216\hfill & =\hfill & 0\hfill & \hfill & \text{Try}\text{\hspace{0.17em}}\text{factoring}\text{.}\hfill \\ \hfill & \hfill & (x-8)(x-27)\hfill & =\hfill & 0\hfill & \hfill & \hfill \\ \hfill & \hfill & x-8=0\hfill & or\hfill & \hfill & x-27=0\hfill & \hfill \\ \hfill & \hfill & x=8\hfill & or\hfill & \hfill & x=27\hfill & \hfill \end{array}\hfill \\ \begin{array}{ccc}& & \text{Check}\text{\hspace{0.17em}}\text{these}\text{\hspace{0.17em}}\text{potential}\text{\hspace{0.17em}}\text{solutions}\text{.}\end{array}\hfill \\ \begin{array}{llllllllll}\text{Step\hspace{0.17em}}4:\hfill & \hfill & \text{If}\text{\hspace{0.17em}}x=8,\hfill & \hfill & \hfill & \hfill & \text{If}\text{\hspace{0.17em}}x=27,\hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill 35\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}8-{8}^{2}& =\hfill & 216\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill 35\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}27-{27}^{2}& =\hfill & 216\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 280-64& =\hfill & 216\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill 945-729& =\hfill & 216\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 216& =\hfill & 216\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill & \hfill 216& =\hfill & 216\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill \end{array}\hfill \\ \begin{array}{ccc}& & \text{These}\text{\hspace{0.17em}}\text{solutions}\text{\hspace{0.17em}}\text{check}\text{.}\end{array}\hfill \\ \hfill \begin{array}{ccc}\text{Step}\text{\hspace{0.17em}}5:& & \text{The}\text{\hspace{0.17em}}\text{computer}\text{\hspace{0.17em}}\text{mouse}\text{\hspace{0.17em}}\text{covers}\text{\hspace{0.17em}}\text{can}\text{\hspace{0.17em}}\text{be}\text{\hspace{0.17em}}\text{priced}\text{\hspace{0.17em}}\text{at}\text{\hspace{0.17em}}\text{either}\text{\hspace{0.17em}}\text{\$8}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\$27}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{order}\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}\text{sell}\text{\hspace{0.17em}}\text{216}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{them}\text{.}\text{\hspace{0.17em}}\end{array}\end{array}$$
A manufacturer of cloth personal computer dust covers notices that the number
$N$ of covers sold is related to the price of covers by
$N=30x-{x}^{2}.$ At what price should the manufacturer price the covers in order to sell 216 of them?
Step 1:
Step 2:
Step 3:
Step 4:
Step 5: In order to sell 216 covers, the manufacturer should price them at either
12 or 18
It is estimated that
$t$ years from now the population of a particular city will be
$P={t}^{2}-24t+96,000.$
How many years from now will the population be 95,865?
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
In 9 and 15 years, the population of the city will be 95,865.
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