Lesson 9 Summing Up Sand Castles Solidify Understanding

Ready

The height of an arrow shot upward from the earth’s surface at an initial velocity of $40 \frac{m}{s}$ can be modeled by the function $f (t) = - 5 t^{2} + 40 t$ .

1.

Calculate the number of seconds the arrow will be in the air.

2.

If you used completing the square on $f (t)$ to find the elapsed time in problem 1, your work would have looked something like this:

$\begin{aligned} 0 & = - 5 t^{2} + 40 t \\ 0 & = - 5 (t^{2} - 8 t + \underset{―}{}) \\ 0 & = - 5 (t^{2} - 8 t + 16) + 5 (16) \\ 0 & = - 5 {(t - 4)}^{2} + 80 \\ - 80 & = - 5 {(t - 4)}^{2} \\ \frac{- 80}{- 5} & = {(t - 4)}^{2} \\ \sqrt{16} & = \sqrt{{(t - 4)}^{2}} \\ \pm 4 & = t - 4 \\ t & = 0, 8 \end{aligned}$

Use $f (t) = - 5 {(t - 4)}^{2} + 80$ to explain why the maximum height of the arrow is $80$ meters and the height was attained at $4$ seconds.

Use the following information for problems 3–5.

The height of an arrow shot upward from the earth’s surface at an initial velocity of $v \frac{m}{s}$ can be modeled by the function $f (t) = - 5 t^{2} + v t$ .

Calculate the number of seconds the arrow will be in the air for the given velocity.
Use completing the square to find the maximum height of the arrow and the time it takes to reach its maximum.

3.

$v = 60 \frac{m}{s}$

a.

Calculate the number of seconds the arrow will be in the air.

b.

Use completing the square to find the maximum height of the arrow and the time it takes to reach its maximum.

4.

$v = 100 \frac{m}{s}$

a.

Calculate the number of seconds the arrow will be in the air.

b.

Use completing the square to find the maximum height of the arrow and the time it takes to reach its maximum.

5.

$v = 250 \frac{m}{s}$

a.

Calculate the number of seconds the arrow will be in the air.

b.

Use completing the square to find the maximum height of the arrow and the time it takes to reach its maximum.

Set

For each of the following:

Write out the sum of terms represented by the notation.
Use the formula you derived in class today for the sum of a geometric series to find the total of the sum of terms without adding each term individually.