A rock is tossed straight up into the air. The height h (in feet) t seconds after it is tossed is given by

the function:

h(t)=-16t^2+100.4t+5.1

(a) Make a table of values for 0 to 10 seconds.
(b) Graph the function.
(c) What is the initial height of the rock?
(d) When did the rock reach its maximum height? What is that height?
(e) At what time did the rock hit the ground?

No one, like no one, uses feet any more for physics problems but anyway

It is a parabola (uppside down, sheds water)

c) at t = 0, h = 5,1

d) find the vertex by completing the square or know that the rock is at the top when the velocity is zero which is when
-32 t + 100.4 = 0
t = 3.14 seconds to top
h = -16(3.14)^2 +100.4(3.14) + 5.1
= -158 + 315 +5.1 = 162 feet

e) solve quadratic for roots and use positive t
3.14+/-3.19
use t = 6.33 seconds

(a) To make a table of values for 0 to 10 seconds, we will substitute different values of t into the function h(t) = -16t^2+100.4t+5.1 and calculate the corresponding height h(t).

Let's substitute values of t from 0 to 10 into the function:

For t = 0:
h(0) = -16(0)^2 + 100.4(0) + 5.1 = 5.1
So, when t = 0, the height is 5.1 feet.

For t = 1:
h(1) = -16(1)^2 + 100.4(1) + 5.1 = 89.5
So, when t = 1, the height is 89.5 feet.

For t = 2:
h(2) = -16(2)^2 + 100.4(2) + 5.1 = 165.1
So, when t = 2, the height is 165.1 feet.

Continue substituting values of t up to 10 to complete the table.

(b) To graph the function h(t) = -16t^2+100.4t+5.1, we can use a graphing calculator, a software like Microsoft Excel, or an online graphing tool. Plot the values from the table (t on the x-axis and h(t) on the y-axis) and join the points to create a smooth curve.

(c) The initial height of the rock is given when t = 0. From the table, we found that h(0) = 5.1 feet. Therefore, the initial height of the rock is 5.1 feet.

(d) To find when the rock reaches its maximum height, we need to determine the vertex of the parabolic function h(t) = -16t^2 + 100.4t + 5.1. The t-coordinate of the vertex is given by the formula t = -b/2a, where the quadratic function is in the form ax^2 + bx + c.

Here, a = -16 and b = 100.4. Substituting these values into the formula, we get:
t = -(100.4) / (2(-16))
t = -100.4 / -32
t = 3.1375

Therefore, the rock reaches its maximum height at approximately t = 3.1375 seconds.

To find the maximum height, substitute this value of t back into the function: h(3.1375) = -16(3.1375)^2 + 100.4(3.1375) + 5.1. Calculate this expression to find the maximum height of the rock.

(e) The rock hits the ground when its height, h(t), is equal to 0. So we need to solve the quadratic equation h(t) = -16t^2+100.4t+5.1 = 0 for t. This can be done by factoring, completing the square, or using the quadratic formula. After solving the equation, you can find the time at which the rock hits the ground.