# To determine: The number of luxury home sales S(t) in a major Canadian urban area over a period of 12 year is given by: Rightarrow S(t)=5.8 t^{2} - 81.2 t + 1200 Question
Modeling data distributions To determine: The number of luxury home sales S(t) in a major Canadian urban area over a period of 12 year is given by: $$\displaystyle\Rightarrow\ {S}{\left({t}\right)}={5.8}\ {t}^{{{2}}}\ -\ {81.2}\ {t}\ +\ {1200}$$ 2021-03-06
Step 1 For minimum number of sales: $$\displaystyle\Rightarrow\ {\frac{{{d}{S}{\left({t}\right)}}}{{{\left.{d}{t}\right.}}}}={0}$$
$$\displaystyle\Rightarrow\ {\frac{{{d}{\left({5.8}{t}^{{{2}}}\ -\ {81.2}{t}\ +\ {1200}\right)}}}{{{\left.{d}{t}\right.}}}}={0}$$
$$\displaystyle\Rightarrow\ {11.6}{t}\ -\ {81.2}={0}$$
$$\displaystyle\Rightarrow\ {11.6}{t}={81.2}$$
$$\displaystyle\Rightarrow\ {t}={\frac{{{81.2}}}{{{11.6}}}}$$
$$\displaystyle\Rightarrow\ {t}={7}$$ Step 2 So, minimum number of sales is given by $$\displaystyle\Rightarrow\ {S}{\left({7}\right)}={5.8}{\left({7}\right)}^{{{2}}}\ -\ {81.2}{\left({7}\right)}\ +\ {1200}$$
$$\displaystyle\Rightarrow\ {S}{\left({7}\right)}={284.2}\ -\ {568.4}\ +\ {1200}$$
$$\displaystyle\Rightarrow\ {S}{\left({7}\right)}={915.8}$$
$$\displaystyle\Rightarrow\ {S}{\left({7}\right)}\ \stackrel{\sim}{=}\ {916}$$

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