Question

The height of a football, in feet, t seconds after it is kicked into the air off of a tee, is modeled by the function h(t) = –16t2 + 35t + 0.25. What is the approximate maximum height of the football?

Answer 1

The maximum height of the football is approximately 18.82 feet.

Step-by-step explanation:

The maximum height of the football can be found by identifying the vertex of the quadratic function. In this case, the function h(t) = -16t^2 + 35t + 0.25 represents the height of the football at time t.

The vertex of a quadratic function occurs at the maximum or minimum point of the graph.

Since the coefficient of the t^2 term is negative, the graph of the function opens downward, indicating a maximum height.

To find the t-coordinate of the vertex, we can use the formula -b/(2a), where a and b are the coefficients of the t^2 and t terms, respectively. In this case, a = -16 and b = 35. Plugging these values into the formula, we get t = -35/(2*-16) ≈ 1.09 seconds.

To find the maximum height, we can substitute this value of t into the function:

h(1.09) = -16(1.09)^2 + 35(1.09) + 0.25 ≈ 18.82 feet.

Therefore, the approximate maximum height of the football is 18.82 feet.

Answer 2

The function is:
f ( t ) = - 16 t² + 35 t + 0.25
We can use the derivative:
f ` ( t ) = - 32 t + 35
- 32 t + 35 = 0
t = 35 : 32 = 1.09375 s
The maximum height is:
f ( 1.09375 ) = - 16 · ( 1.09357 )² + 35 · 1.09357 + 0.25 =
= - 19.14 + 38.28125 + 0.25 = 19.39 ft ≈ 19.4 ft.
Answer: The approximate maximum height of the ball is 19.4 ft.