A juggler is performing her act using several balls. She throws the balls up at an initial height of 4 feet, with a speed of 15 feet per second. This can be represented by the function [tex]H(t) = -16t^2 + 15t + 4[/tex]. If the juggler doesn't catch one of the balls, about how long does it take the ball to hit the floor?

A. 7.52 seconds
B. 1.15 seconds
C. 0.47 seconds
D. 0.22 seconds



Answer :

To determine the time it takes for the ball to hit the floor, we need to find the value of \( t \) when the height \( H(t) \) equals zero. The height can be modeled by the quadratic equation:

[tex]\[ H(t) = -16t^2 + 15t + 4 \][/tex]

We set \( H(t) = 0 \):

[tex]\[ -16t^2 + 15t + 4 = 0 \][/tex]

This is a quadratic equation of the form \( at^2 + bt + c = 0 \), where:
- \( a = -16 \)
- \( b = 15 \)
- \( c = 4 \)

To solve for \( t \), we use the quadratic formula:

[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

First, we compute the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]

Substitute \( a = -16 \), \( b = 15 \), and \( c = 4 \) into the discriminant formula:
[tex]\[ \text{Discriminant} = 15^2 - 4(-16)(4) \][/tex]
[tex]\[ \text{Discriminant} = 225 + 256 \][/tex]
[tex]\[ \text{Sciminant} = 481 \][/tex]

Next, we substitute the values of \( a \), \( b \), and the discriminant back into the quadratic formula:

[tex]\[ t = \frac{-15 \pm \sqrt{481}}{2(-16)} \][/tex]

Calculating both potential solutions:

[tex]\[ t_1 = \frac{-15 + \sqrt{481}}{-32} \][/tex]
[tex]\[ t_2 = \frac{-15 - \sqrt{481}}{-32} \][/tex]

These yield the approximate solutions:

[tex]\[ t_1 \approx -0.217 \][/tex]
[tex]\[ t_2 \approx 1.154 \][/tex]

Given that time cannot be negative, we discard the negative solution. Therefore, the time it takes for the ball to hit the floor is approximately:

[tex]\[ t \approx 1.154 \][/tex]

So the closest answer to the options provided is:

1.15 seconds