Question

1. a baseball is thrown from a height of 5 feet. the height, h, of the ball at time t seconds is modeled by the equation h(t) = -16t² + 100t + 5. how long will it take the ball to reach the ground? (you may use calculator)

Explanation:

Step1: Set \( h(t) = 0 \)

We need to find the time \( t \) when the ball reaches the ground, so we set the height function \( h(t) = 0 \). This gives us the quadratic equation:

$$ -16t^{2}+100t + 5=0 $$

Step2: Use quadratic formula

For a quadratic equation of the form \( ax^{2}+bx + c = 0 \), the quadratic formula is \( t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \). In our equation, \( a=- 16 \), \( b = 100 \), and \( c = 5 \). First, we calculate the discriminant \( D=b^{2}-4ac \):

$$ D=(100)^{2}-4\times(-16)\times5=10000 + 320=10320 $$

Then we find the square root of the discriminant: \( \sqrt{10320}\approx101.587 \)
Now we substitute \( a \), \( b \), and \( D \) into the quadratic formula:

$$ t=\frac{-100\pm101.587}{2\times(-16)} $$

We have two solutions:

• For the plus sign: \( t=\frac{-100 + 101.587}{-32}=\frac{1.587}{-32}\approx - 0.0496 \) (we discard this negative solution since time cannot be negative)
• For the minus sign: \( t=\frac{-100-101.587}{-32}=\frac{-201.587}{-32}\approx6.3 \)

Answer:

Approximately \( 6.3 \) seconds (the more precise value can be calculated as \( t=\frac{-100-\sqrt{10320}}{-32}=\frac{100 + \sqrt{10320}}{32}\approx\frac{100+101.587}{32}=\frac{201.587}{32}\approx6.3 \))