Question

5.5 practice problem
the equation $h(t) = -16t^2 + 80t + 64$ represents the height, in feet, of a potato $t$ seconds after it was launched from a mechanical device.

9 fill in the blank 1 point
question 8a
write an equation that would allow us to find the time the potato hits the ground. $0 = $

10 essay 1 point
question 8b
solve the equation without graphing. show your reasoning.

Explanation:

Step1: Set height to 0 (ground level)

$0 = -16t^2 + 80t + 64$

Step2: Simplify the equation

Divide all terms by $-16$:
$t^2 - 5t - 4 = 0$

Step3: Apply quadratic formula

For $at^2+bt+c=0$, $t=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Here $a=1$, $b=-5$, $c=-4$:
$t=\frac{5\pm\sqrt{(-5)^2-4(1)(-4)}}{2(1)}=\frac{5\pm\sqrt{25+16}}{2}=\frac{5\pm\sqrt{41}}{2}$

Step4: Select positive time value

$\frac{5-\sqrt{41}}{2}$ is negative, so discard. Calculate positive root:
$t=\frac{5+\sqrt{41}}{2}$

Answer:

Question 8a:

$0 = -16t^2 + 80t + 64$

Question 8b:

$t=\frac{5+\sqrt{41}}{2}$ seconds (or approximately 5.70 seconds)