Question

4 Ellie is in a snowboarding competition. The function (t) =- 16t² + 30t + 10 models Ellie's height, & in the air afterseconds. Use the quadratic formula to determine, to the nearest tenth of a second, at what two times her height was 15 feet.Show all steps

Answer 1

Given: Ellie's height function

$\begin{gathered} H(t)=-16t^2+30t+10 \\ H(t)=Height \\ t=time \end{gathered}$

To Determine: The two times when the height was 15 feet

Solution

Given the quadratic formula below

$\begin{gathered} A\text{ quadratic equation below} \\ ax^2+bx+c=0 \\ Using\text{ quadratic formula} \\ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \end{gathered}$

When height is 15 feet

$\begin{gathered} 15=-16t^2+30t+10 \\ 16t^2-30t-10+15=0 \\ 16t^2-30t+5=0 \end{gathered}$

Using quadratic formula

$\begin{gathered} a=16,b=-30,c=5 \\ t=\frac{-(-30)\pm\sqrt{(-30)^2-4\times16\times5}}{2\times16} \end{gathered}$$t=\frac{-\left(-30\right)\pm\:2\sqrt{145}}{2\cdot\:16}$$t_1=\frac{-\left(-30\right)+2\sqrt{145}}{2\cdot \:16},\:t_2=\frac{-\left(-30\right)-2\sqrt{145}}{2\cdot \:16}$$\begin{gathered} t_1=\frac{30+24.083}{32},t_2=\frac{30-24.083}{32} \\ t_1=\frac{54.083}{32},t_2=\frac{6.083}{32} \end{gathered}$$\begin{gathered} t_1=1.69,t_2=0.19 \\ t_1\approx1.7seconds,t_2=0.2seconds \end{gathered}$

Hence, the two times Ellies height was 25 feet is 1.7 seconds and 0.2 seconds