During halftime of a football​game, a sling shot launches​ T-shirts at the crowd. A​ T-shirt is launched from a height of 33 feet with an initial upward velocity of 64 feet per second. The​ T-shirt is caught 48 feet above the field. How long will it take the​ T-shirt to reach its maximum​ height? What is the maximum​ height? What is the range of the function that models the height of the​ T-shirt over​ time?

2.25 to get to the sky

84 is the max
3<_h<_84

Well, let me calculate that for you while adding a pinch of humor. Let's call this the "Flying T-Shirt Show".

To find out how long it takes for the T-shirt to reach its maximum height, we can use the formula for the time it takes to reach maximum height in a projectile motion:

time = (initial vertical velocity) / (acceleration due to gravity)

But hey, let's not complicate things! In this case, we know the T-shirt was shot with an initial upward velocity of 64 feet per second. So, by dividing that by the acceleration due to gravity (which is about 32 feet per second squared), we find it takes a whopping 2 seconds for that T-shirt to reach its peak!

Now, the maximum height. Well, assuming there are no strong gusts of wind or mischievous seagulls at play, we can calculate the maximum height by using the formula for vertical displacement:

max height = initial height + (initial vertical velocity^2) / (2 * acceleration due to gravity)

Plugging in the values, we get:

max height = 33 feet + (64 feet per second^2) / (2 * 32 feet per second^2)

Calculating this marvelous feat, we find that the T-shirt reaches a mind-blowing maximum height of 65.5 feet! It's almost like it's trying to touch the stars.

Finally, the range of the function that models the height over time. Well, the range refers to the set of all possible output values. In this case, the range of the function would be from the lowest height (0 feet, when it's caught) to the highest height (the maximum height of 65.5 feet). So, the range of the function is from 0 to 65.5 feet, capturing the full glory of the T-shirt's flight through the air.

And there you have it, my friend! The grand tale of the T-shirt's journey, complete with humor and mathematical prowess.

To find out how long it will take for the T-shirt to reach its maximum height, we can use the formula for the time it takes for an object to reach its peak height in projectile motion:

t = v0 / g

where:
t = time taken
v0 = initial velocity
g = acceleration due to gravity (approximately 32.2 ft/s^2)

In this case, the initial velocity is 64 feet per second, so:

t = 64 ft/s / 32.2 ft/s^2
t ≈ 1.99 seconds

Therefore, it will take approximately 1.99 seconds for the T-shirt to reach its maximum height.

To calculate the maximum height, we can use the formula for the peak height in projectile motion:

h_max = h0 + (v0^2 / (2 * g))

where:
h_max = maximum height
h0 = initial height
v0 = initial velocity
g = acceleration due to gravity (approximately 32.2 ft/s^2)

In this case, the initial height is 33 feet and the initial velocity is 64 feet per second, so:

h_max = 33 ft + (64 ft/s)^2 / (2 * 32.2 ft/s^2)
h_max ≈ 33 ft + 2048 ft^2/s^2 / 64.4 ft/s^2
h_max ≈ 33 ft + 31.75 ft
h_max ≈ 64.75 ft

Therefore, the maximum height reached by the T-shirt is approximately 64.75 feet.

The range of the function that models the height of the T-shirt over time is the horizontal distance traveled by the T-shirt from the starting point to the point where it lands. The range can be calculated using the formula:

R = v0 * t

where:
R = range
v0 = initial velocity
t = time taken

In this case, the initial velocity is 64 feet per second and the time taken is approximately 1.99 seconds, so:

R = 64 ft/s * 1.99 s
R ≈ 127.36 ft

Therefore, the range of the function that models the height of the T-shirt over time is approximately 127.36 feet.

well, the height y at time t is

y = 33 + 64t - 16t^2

So, solve for t when y=48.
The max height is of course, when t = 64/32 = 2.

I expect you can determine the range... (restrict it to the domain involved.)

During halftime of a football ​game, a sling shot launches​ T-shirts at the crowd. A​ T-shirt is launched from a height of 5 feet with an initial upward velocity of 64 feet per second. The​ T-shirt is caught 47 feet above the field. How long will it take the​ T-shirt to reach its maximum​ height? What is the maximum​ height? What is the range of the function that models the height of the​ T-shirt over​ time? can you solve for max height and the amount of seconds it take to reach max height