A certain baseball hit straight up in the air, is at height represented by:

h(t) = 4 + 50t – 16t2 , where t is time in seconds after being hit and h(t) is in feet. The function is valid for t > 0, until the ball hits the ground.

a) Sketch the graph of this function.
b) How high is it at t = 0?
c) When does it hit the ground?
d) When, if ever, is it 30 feet high?
e) When, if ever, is it 90 feet high?
f) When does it reach the maximum height?
g) What is the maximum height that it reaches?

a) use fooplot dot com

b) 4 ft

c) set h=0 , then solve for t

d) set h=30 , then solve for t
... two solutions

e) never

f) axis of sym ... 50 / (2 * 16)

g) use value from f) in h(t)

To answer these questions, we will analyze the given function h(t) = 4 + 50t - 16t^2.

a) Sketching the graph of the function:
To sketch the graph, we need to plot points on a coordinate plane using the function. We can choose various values of t and substitute them into the function to find the corresponding heights h(t).

b) Height at t = 0:
To find the height at t = 0, we substitute t = 0 into the function h(t). So, h(0) = 4 + 50(0) -16(0)^2 = 4. Therefore, the height at t = 0 is 4 feet.

c) When it hits the ground:
The ball hits the ground when its height is zero. So, we need to solve the equation h(t) = 0 for t. Setting h(t) = 0 in the function, we get 4 + 50t - 16t^2 = 0. This is a quadratic equation that can be solved using factoring, completing the square, or the quadratic formula. The solutions would give us the times when the ball hits the ground.

d) When it is 30 feet high:
We need to solve the equation h(t) = 30 for t. So, we set 4 + 50t - 16t^2 = 30 and solve for t.

e) When it is 90 feet high:
We need to solve the equation h(t) = 90 for t. So, we set 4 + 50t - 16t^2 = 90 and solve for t.

f) When it reaches the maximum height:
The maximum height is reached at the vertex of the parabola. To find the t-coordinate of the vertex, we can use the formula t = -b/(2a), where a, b, and c are the coefficients of the quadratic equation in standard form. In this case, a = -16, b = 50. The t-value at the vertex will give us the time when the maximum height is reached.

g) Maximum height:
To find the maximum height, we substitute the t-coordinate of the vertex into the function h(t) and evaluate h(t).

By following these steps, you can find the answers to all the given questions related to this function.