During halftime of a soccer game, a sling shot launched t-shirts at the crowd. a t-shirt is launched from a height of 3 feet with an initial upward velocity of 80 feet per second. the t-shirt is caught 43 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?

Step-by-step explanation:

Ballistic motion is usually modeled by the quadratic function ...

  h(t) = -1/2g·t^2 +v0·t +h0

where g is the acceleration due to gravity, v0 is the initial upward velocity, and h0 is the initial height.

When units are in feet and seconds, the value of g is usually taken to be 32 ft/s^2, so for v0 = 80 ft/s and h0 = 3 ft, the equation is ...

  h(t) = -16t^2 +80t +3

The axis of symmetry of the graph of this equation is ...

  t = v0/g = (80 ft/s)/(32 ft/s^2)

  t = 2.5 s . . . . . the time to reach the maximum height

At that time, the maximum height is ...

  h(2.5) = (-16·2.5 +80)2.5 +3 = 103 . . . . feet

Then the range of the function extends from its minimum value, 3 at t=0, to its maximum value, 103 at t=2.5. Since the T-shirt is caught at h(t) = 43, the T-shirt never goes below 3 feet.

The range of the function is 3 ft to 103 ft.

_____

Alternate solution

The graph suggests an alternate solution. Since the T-shirt is caught at a height of 43 ft, this could be the maximum height the T-shirt reaches. That will occur at about t=0.564 seconds. Then the range of the function is 3–43 feet.

answer: this is not a question

step-by-step explanation: this is a answer

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240