Question

a model rocket is launched with an initial upward velocity of 164 ft/s. the rockets height h (in feet) after t seconds is given by the following. h = 164t - 16t². find all values of t for which the rockets height is 92 feet. round your answer(s) to the nearest hundredth. (if there is more than one answer, use the \or\ button.)

Explanation:

Step1: Set up the equation

Set $h = 92$ in the equation $h=164t - 16t^{2}$, so we get $92=164t - 16t^{2}$. Rearrange it to the standard - form of a quadratic equation $16t^{2}-164t + 92 = 0$. Divide through by 4 to simplify: $4t^{2}-41t + 23 = 0$.

Step2: Use the quadratic formula

For a quadratic equation $ax^{2}+bx + c = 0$ ($a = 4$, $b=-41$, $c = 23$), the quadratic formula is $t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. First, calculate the discriminant $\Delta=b^{2}-4ac=(-41)^{2}-4\times4\times23=1681 - 368 = 1313$.

Step3: Find the values of t

$t=\frac{41\pm\sqrt{1313}}{8}$. $\sqrt{1313}\approx36.235$. Then $t_1=\frac{41 + 36.235}{8}=\frac{77.235}{8}\approx9.65$ and $t_2=\frac{41-36.235}{8}=\frac{4.765}{8}\approx0.60$.

Answer:

$t\approx0.60$ or $t\approx9.65$