Use the quadratic formula on the function h=−16t2+80t+3

Question

ℎ
=
−
16
𝑡
2
+
80
𝑡
+
3
to calculate the time it will take for the baseball to hit the ground. Round to the nearest hundredth, if necessary.

Reference Sheet - Quadratic Formula

(1 point)
The baseball will hit the ground at about seconds.

Answer 1

To find the time $t$ when the baseball hits the ground, we need to set the height $h$ to zero and solve the quadratic equation:

$$-16t^2+80t+3=0$$

We can use the quadratic formula, which is given by:

$$t=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

In the equation, $a=-16$, $b=80$, and $c=3$. Let's calculate the discriminant $b^2-4ac$:

1. Calculate $b^2$:

$$b^2={80}^2=6400$$

2. Calculate $4ac$:

$$4ac=4\cdot (-16)\cdot 3=-192$$

3. Now calculate the discriminant:

$$b^2-4ac=6400-(-192)=6400+192=6592$$

Now we can use the quadratic formula:

$$t=\frac{-80\pm \sqrt{6592}}{2\cdot (-16)}$$

Let's compute $\sqrt{6592}$:

$$\sqrt{6592}\approx 81.1\text{(approximately)}$$

Now, substituting back into the formula:

$$t=\frac{-80\pm 81.1}{-32}$$

This gives us two potential solutions:

1. Using the plus sign:

$$t_1=\frac{-80+81.1}{-32}=\frac{1.1}{-32}\approx -0.034375\text{(not a valid solution, as time cannot be negative)}$$

2. Using the minus sign:

$$t_2=\frac{-80-81.1}{-32}=\frac{-161.1}{-32}\approx 5.03$$

So rounding to the nearest hundredth:

$$t\approx 5.03\text{ seconds}$$

Therefore, the baseball will hit the ground at about 5.03 seconds.