A child throws a penny upward out of a window and watches it fall to the ground. The function below represents the penny's distance in feet above the ground f seconds after the penny is thrown.

Question

Approximately how many seconds does it take the penny to hit the ground?

f(t) = -16t^2 + t + 10(1 point)
Responses

0.03

0.76

0.82

1.32

Answer 1

To find out how many seconds it takes for the penny to hit the ground, we need to determine when the distance above the ground \( f(t) \) is equal to 0. The function given is:

\[ f(t) = -16t^2 + t + 10 \]

We set \( f(t) = 0 \) to find when the penny hits the ground:

\[ -16t^2 + t + 10 = 0 \]

We can use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -16 \), \( b = 1 \), and \( c = 10 \).

Calculating the discriminant:

\[ b^2 - 4ac = 1^2 - 4(-16)(10) = 1 + 640 = 641 \]

Now using the quadratic formula:

\[ t = \frac{-1 \pm \sqrt{641}}{2(-16)} \]

Calculating \( \sqrt{641} \):

\[ \sqrt{641} \approx 25.31 \]

Now we can plug it into the formula:

\[ t = \frac{-1 \pm 25.31}{-32} \]

Calculating the two possible times:

1. Using the positive root:

\[ t_1 = \frac{-1 + 25.31}{-32} = \frac{24.31}{-32} \approx -0.76 \quad \text{(not valid)} \]

2. Using the negative root:

\[ t_2 = \frac{-1 - 25.31}{-32} = \frac{-26.31}{-32} \approx 0.82 \]

Thus, the time it takes for the penny to hit the ground is approximately 0.82 seconds. Therefore, the answer is:

0.82