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 \( \sqrt{6592} \):

\[ \sqrt{6592} \approx 81.1 \quad \text{(approximately)} \]

Now, substituting back into the formula:

\[ t = \frac{-80 \pm 81.1}{-32} \]

This gives us two potential solutions:

1. Using the plus sign:

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

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.