in the past professional baseball was played at the astrodome in Houston, Texas. the astrodome has a maximum height of 63.4 m. the height of a baseball t seconds after it is hit straight up in the air with a velocity of 45 ft/s is given by h = -9.8t + 45t + 1. will a baseball hit straight up with this velocity hit the roof of the astrodome? use the discriminant to explain your answer.

Question

in the past professional baseball was played at the astrodome in Houston, Texas. the astrodome has a maximum height of 63.4 m. the height of a baseball t seconds after it is hit straight up in the air with a velocity of 45 ft/s is given by h = -9.8t + 45t + 1. will a baseball hit straight up with this velocity hit the roof of the astrodome? use the discriminant to explain your answer.

Answer 1

V = Vo + g*Tr

Tr = (V-Vo)/g = (0-45)/-32 = 1.41 s. =
Rise time or time to reach max. ht.

hmax = -16t^2 + 45t + 1
hmax=-16(1.41)^2 + 45*1.41 + 1=32.64 Ft.
= 9.89 m. Will not hit roof.

Note: Your Eq is not correct.

Answer 2

To determine if the baseball will hit the roof of the Astrodome, we need to find the time it takes for the baseball to reach a height of 63.4 m.

Given the equation h = -9.8t^2 + 45t + 1, where h represents the height and t represents time, we can set this equation equal to 63.4 and solve for t:

-9.8t^2 + 45t + 1 = 63.4

Simplifying the equation, we have:

-9.8t^2 + 45t - 62.4 = 0

Now, let's calculate the discriminant, which is the part of the quadratic formula under the square root:

D = b^2 - 4ac

In this case, a = -9.8, b = 45, and c = -62.4. Substituting these values, we have:

D = (45)^2 - 4(-9.8)(-62.4)

Simplifying further:

D = 2025 + 2450.4

D = 4475.4

Since the discriminant is positive (D > 0), it indicates that there are two real solutions for the time at which the ball reaches a height of 63.4 m. This means the ball will reach that height twice.

However, the discriminant does not tell us whether the ball will hit the roof of the Astrodome. To determine that, we have to examine the actual values of t.

Solving the quadratic equation, we find two values of t:

t = (-b ± √D) / 2a

t = (-45 ± √(4475.4)) / (2 * -9.8)

Using a calculator to find the square root and evaluate the equation, we get:

t = (-45 ± 66.92) / (-19.6)

This gives us two values of t:

t1 ≈ 0.3 seconds
t2 ≈ 4.1 seconds

Since the time it takes for the baseball to reach a height of 63.4 m is less than 4.1 seconds, the baseball will indeed hit the roof of the Astrodome.