Question

a model rocket is launched with an initial upward velocity of 215 ft/s. the rockets height h (in feet) after t seconds is given by the following. h = 215t - 16t^2 find all values of t for which the rockets height is 97 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 $h = 97$, so $97=215t - 16t^{2}$. Rearrange to get $16t^{2}-215t + 97=0$.

Step2: Identify coefficients

For the quadratic equation $at^{2}+bt + c = 0$, here $a = 16$, $b=-215$, $c = 97$.

Step3: Apply quadratic formula

The quadratic formula is $t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Substitute the values: $t=\frac{215\pm\sqrt{(-215)^{2}-4\times16\times97}}{2\times16}$.

Step4: Calculate discriminant

First, calculate the discriminant $\Delta=b^{2}-4ac=(-215)^{2}-4\times16\times97=46225 - 6208=40017$.

Step5: Find values of t

$t=\frac{215\pm\sqrt{40017}}{32}$. $\sqrt{40017}\approx200.0425$. So $t_1=\frac{215 + 200.0425}{32}\approx\frac{415.0425}{32}\approx12.97$ and $t_2=\frac{215-200.0425}{32}\approx\frac{14.9575}{32}\approx0.47$.

Answer:

$t\approx0.47$ or $t\approx12.97$