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
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asked 2021-01-13
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.

Answers (1)

2021-01-14

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, \(\displaystyle{t}={\frac{{{3}}}{{{2}}}}\)

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