xoxluna
contestada

A model rocket is launched with an initial upward velocity of 100 ft/s. The rocket's height h (in feet) after t seconds is given by the following.

h=100t-16t^2

Find all values of t for which the rocket's height is 34 feet.
Round your answer(s) to the nearest hundredth.

Respuesta :

[tex]\bf \stackrel{h}{34}=100t-16t^2\implies 16t^2-100t+34=0\\\\\\ 8t^2-50t+17=0 \\\\\\ ~~~~~~~~~~~~\textit{quadratic formula} \\\\ \stackrel{\stackrel{a}{\downarrow }}{8}t^2\stackrel{\stackrel{b}{\downarrow }}{-50}t\stackrel{\stackrel{c}{\downarrow }}{+17}=0 \qquad \qquad t= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a}[/tex]

[tex]\bf t=\cfrac{ - (-50) \pm \sqrt { (-50)^2 -4(8)(17)}}{2(8)}\implies t=\cfrac{50\pm \sqrt{2500-544}}{16} \\\\\\ t=\cfrac{50\pm\sqrt{1956}}{16}\implies t=\cfrac{50\pm\sqrt{4\cdot 489}}{16}\implies t=\cfrac{50\pm\sqrt{2^2\cdot 489}}{16} \\\\\\ t=\cfrac{50\pm 2\sqrt{489}}{16}\implies t=\cfrac{25\pm \sqrt{489}}{8}\implies t\approx \begin{cases} 5.88916805\\ 0.36083195 \end{cases}[/tex]

The value of t for which the rocket's height is 34 feet

[tex]$ t \approx\left\{\begin{array}{l}5.88916805 \\ 0.36083195\end{array}\right.$[/tex]

Quadratic equation formula

The value of t for which the rocket's height is 34 feet.

[tex]h=100t-16t^2[/tex]

Where, h = 34

[tex]${34}=100 \mathrm{t}-16 \mathrm{t}^{2}[/tex]

[tex]\Longrightarrow 16 \mathrm{t}^{2}-100 \mathrm{t}+34=0$[/tex]

[tex]$8 t^{2}-50 t+17=0$[/tex]

By using quadratic equation formula, we get

Let a = 8, b = -50 and c = 17

[tex]{8} t^{2}-50 t+17=0[/tex]

Substituting the values of a, b, and c in the above equation, we get

[tex]$t=\frac{-(-50) \pm \sqrt{(-50)^{2}-4(8)(17)}}{2(8)}[/tex]

[tex]$\Longrightarrow t=\frac{50 \pm \sqrt{2500-544}}{16}$[/tex]

[tex]$\mathrm{t}=\frac{50 \pm \sqrt{1956}}{16}[/tex]

Separate the solutions

[tex]$\Longrightarrow \mathrm{t}=\frac{50 \pm \sqrt{4 \cdot 489}}{16}[/tex]

[tex]$\Longrightarrow \mathrm{t}=\frac{50 \pm \sqrt{2^{2} \cdot 489}}{16}$[/tex]

[tex]$\mathrm{t}=\frac{50 \pm 2 \sqrt{489}}{16}[/tex]

[tex]$\Longrightarrow \mathrm{t}=\frac{25 \pm \sqrt{489}}{8}[/tex]

then we get

[tex]$\Longrightarrow t \approx\left\{\begin{array}{l}5.88916805 \\ 0.36083195\end{array}\right.$[/tex]

Therefore, the value of t for which the rocket's height is 34 feet

[tex]$ t \approx\left\{\begin{array}{l}5.88916805 \\ 0.36083195\end{array}\right.$[/tex]

To learn more about quadratic formula equation

brainly.com/question/1538323

#SPJ2

Otras preguntas