A projectile is fired straight up in the air. Let h(t) be its height in meters above the ground t seconds after it was fired. Use the model

Question

h(t) = −4.9t2 + 70t + 30
to answer the following questions. Feel free to use a calculator for computations, but show all your work for things like factoring or the quadratic formula. Use function notation when showing your work and answer the questions in words. Give decimal values for all answers, rounding to the nearest hundredth.
(a) What was the height of the projectile 8 seconds after it was fired?
(b) What was the height of the projectile a the moment when it was fired?
(c) At what time will the projectile hit the ground?
(d) What will the maximum height of the projectile be?
(e) Sketch the graph of h(t). Make it big and label all of the information you’ve found so far.
(f) What is the domain of the function?

Answer 1

( a )

Put t = 8 in your formula

h(8) = − 4.9 * 8 ^ 2 + 70 * 8 + 30

h(8) = − 4.9 * 64 + 560 + 30

h(8) = − 4.9 * 64 + 560 + 30

h(8) = − 313.6 + 590

h(8) = 276.4

( b )

Put t = 0 in your formula

h(0) = − 4.9 * 0 ^ 2 + 70 * 0 + 30

h(0) = − 4.9 * 0 + 0 + 30

h(0) = 0 + 30

h(0) = 30

( c )

When the projectile hit the ground then h = 0

So you must solve quadratic eguation:

− 4.9 t ^ 2 + 70 t + 30 = 0

Try that.

The solutions are:

t = ( 10 / 7 ) [ 5 - 2 sqroot ( 7 ) ] = - 0. 41643

and

t = ( 10 / 7 ) [ 5 + 2 sqroot ( 7 ) ] = 14.7021466

Time can't be negative so:

t = 14.70 sec

rounded on two decimal pieces

( d )

The parabola upside-down like in this case has a highest point when:

t = - b / 2 a

This point is called the "vertex".

In this case a = - 4.9 b = 70 c = 30 so:

t = - 70 / 2 * ( - 4.9 )

t = - 70 / - 9.8

t = 700 / 98

t = 2 * 2 * 25 * 7 / ( 2 * 7 * 7 )

t = 2 * 25 / 7

t = 50 / 7

t = 7.142857

t = 7.14 sec

rounded on two decimal pieces

Replace t = 50 / 7 in your equation.

hmax = t ( 50 / 7 )

hmax = - 4.9 * ( 50 / 7 ) ^ 2 + 70 * 50 / 7 + 30 =

- 4.9 * 250 / 49 + 7 * 10 * 50 / 7 + 30 =

- 4.9 * 2500 / ( 4.9 * 10 ) + 10 * 50 + 30 =

- 2500 / 10 + 500 + 30 =

- 250 + 500 + 30 =

250 + 30 = 280

( e )

Go on:

wolframalpha. c o m

When page be open in rectangle type:

plot − 4.9 t ^ 2 + 70 t + 30

Draw diagram on your paper then mark points.

( f )

In mathematical meaning the domain is - infinity to infinity

But time can't be negative and a projectile can't fly after hit the ground, so domain is t [ 0 , 14.70 ]