Questions: After a great deal of experimentation, two college senior physics majors determined that when a bottle of French champagne is shaken several times, held upright, and uncorked, its cork travels according to the function below, where s is its height (in feet) above the ground t seconds after being released. s(t)=-16t^2+30t+4 a. How high will it go? b. How long is it in the air? a. The cork will go feet. (Round to the nearest whole number as needed) b. The cork will be in the air for seconds. (Type an integer or decimal rounded to two decimal places as needed.)

Solution Steps

To find the maximum height the cork will reach, we need to find the vertex of the quadratic function \( s(t) = -16t^2 + 30t + 4 \). The vertex form of a quadratic function \( ax^2 + bx + c \) has its maximum (or minimum) at \( t = -\frac{b}{2a} \).

For our function:

• \( a = -16 \)
• \( b = 30 \)

The time at which the maximum height occurs is: \[ t = -\frac{30}{2 \times -16} = \frac{30}{32} = 0.9375 \text{ seconds} \]

Substitute \( t = 0.9375 \) back into the function to find the maximum height: \[ s(0.9375) = -16(0.9375)^2 + 30(0.9375) + 4 \] \[ s(0.9375) = -16(0.8789) + 28.125 + 4 \] \[ s(0.9375) = -14.0624 + 28.125 + 4 \] \[ s(0.9375) = 18.0626 \text{ feet} \]

Rounding to the nearest whole number, the maximum height is 18 feet.

To find how long the cork is in the air, we need to determine when it hits the ground, i.e., when \( s(t) = 0 \).

Solve the equation: \[ -16t^2 + 30t + 4 = 0 \]

Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

• \( a = -16 \)
• \( b = 30 \)
• \( c = 4 \)

Calculate the discriminant: \[ b^2 - 4ac = 30^2 - 4(-16)(4) = 900 + 256 = 1156 \]

Calculate the roots: \[ t = \frac{-30 \pm \sqrt{1156}}{-32} \] \[ t = \frac{-30 \pm 34}{-32} \]

The two solutions are: \[ t_1 = \frac{4}{-32} = -0.125 \quad (\text{not valid as time cannot be negative}) \] \[ t_2 = \frac{-64}{-32} = 2.0 \]

Thus, the cork is in the air for 2.0 seconds.

Final Answer