Question

The function h= - 16t^{2} + 48t represents the height h (in feet) of a kickball t seconds after it is kicked from the ground. a) Find the maximum height of the kickball. b) Find and interpet the axis of symmetry.

Modeling data distributions
The function $$\displaystyle{h}=\ -{16}{t}^{{{2}}}\ +\ {48}{t}$$ represents the height h (in feet) of a kickball t seconds after it is kicked from the ground. a) Find the maximum height of the kickball. b) Find and interpet the axis of symmetry.

a) Concept used: Maximum or minimum value of the function. Calculation: On comparing the equation $$\displaystyle{a}{x}^{{{2}}}\ +\ {b}{x}\ +\ {c}$$,
$$\displaystyle{a}=\ -{16},\ {b}={48},\ {c}={0}$$ Since, $$\displaystyle{a}\ {<}\ {0}$$ The function has minimum value. For minimum value, Calculate t, $$\displaystyle{t}={\frac{{-{b}}}{{{2}{a}}}}$$ Therefore, $$\displaystyle{t}={\frac{{-{\left({48}\right)}}}{{{2}{\left(-{16}\right)}}}}$$ So, $$\displaystyle{t}={\frac{{-{48}}}{{-{32}}}}$$ On substituting $$\displaystyle{t}={\frac{{{3}}}{{{2}}}}$$ in equation. $$\displaystyle{h}=\ -{16}\ {\left({\frac{{{3}}}{{{2}}}}\right)}^{{{2}}}\ +\ {48}\ {\left({\frac{{{3}}}{{{2}}}}\right)}$$ On squaring, $$\displaystyle{h}=\ -{16}\ {\left({\frac{{{9}}}{{{2}}}}\right)}\ +\ {48}\ {\left({\frac{{{3}}}{{{2}}}}\right)}$$ On solving, $$\displaystyle{h}=\ -{4}\ {\left({9}\right)}\ +\ {24}\ {\left({3}\right)}$$ Therefore, $$\displaystyle{t}={36}$$ b) Concept used: Formula for axis of symmetry, $$\displaystyle{t}={\frac{{-{b}}}{{{2}{a}}}}.$$ Calculation: On comparing the equation $$\displaystyle{a}{x}^{{{2}}}\ +\ {b}{x}\ +\ {c},$$
$$\displaystyle{a}=\ -{16},\ {b}={48},\ {c}={0}$$ Calculate the axis of symmetry, $$\displaystyle{t}={\frac{{-{b}}}{{{2}{a}}}}$$ Therefore, $$\displaystyle{t}={\frac{{-{48}}}{{{2}{\left(-{16}\right)}}}}$$ So, $$\displaystyle{t}={\frac{{-{48}}}{{-{32}}}}$$ The axis of symmetry, $$\displaystyle{t}={\frac{{{3}}}{{{2}}}}$$