Question 1183238
.
Hayden is playing with a toy airplane. The height of the airplane, in metres, is given by the relation
h = −5t^2+ 18t + 1, where t is the number of seconds since the plane was launched. At time t, the
horizontal distance travelled by the airplane is given by d = 4.5t.
a) For how long will the airplane be in the air, rounded to the nearest tenth of a second? 
b) How far will the airplane travel horizontally by the time it hits the ground, rounded to the nearest tenth of a metre? 
c) At what time(s), rounded to the nearest tenth of a second, will the airplane reach a height of 12 m? 
d) What is the maximum height or the plane?
~~~~~~~~~~~~~~



From the equations, the initial vertical speed is 18 m/s up;  the initial height is 1 meter (at the start);

horizontal speed is 4.5 m/s, the constant value during the flight.



a)    For how long will the airplane be in the air, rounded to the nearest tenth of a second? 


<pre>
    The toy airplane is in the air until h(t) > 0.  The airplane get the ground when h(t) = 0.

    THEREFORE, to find the time the airplane is in the air, we should solve this quadratic equation


        -5t^2 + 18t + 1 = 0

        {{{t[1,2]}}} = {{{(-18 +- sqrt(18^2 - 4*(-5)))/(2*(-5))}}} = {{{(-18 +- sqrt(344))/(-10)}}}.


    Of the two roots, we chose the positive one, which is  t = {{{(-18 - sqrt(344))/(-10)}}} = 3.7 seconds  (rounded as requested).   

    So, question (a) is answered: the airplane is in the air 3.7 seconds.
</pre>


b)  &nbsp;&nbsp;How far will the airplane travel horizontally by the time it hits the ground, rounded to the nearest tenth of a metre? 


<pre>
    Simply multiply the horizontal component of the speed by the time

        horizontal distance = 4.5 * 3.7 = 16.45 meters.    <U>ANSWER</U>
</pre>


c)  &nbsp;&nbsp;At what time(s), rounded to the nearest tenth of a second, will the airplane reach a height of 12 m? 


<pre>
    To answer, we should solve this quadratic equation    

        -5t^2 + 18t + 1 = 12

        -5t^2 + 18t - 11 = 0

         {{{t[1,2]}}} = {{{(-18 +- sqrt(18^2 - 4*(-5)*(-11)))/(2*(-5))}}} = {{{(-18 +- sqrt(544))/(-10)}}}.

         {{{t[1]}}} = {{{(-18 +- 10.198)/(-10)}}} = 0.78 s;  {{{t[2]}}} = {{{(-18 - 10.198)/(-10)}}} = 2.82 s.


         The smaller value of t is the ascending flight;  the larger value of t is the descending flight.


         Part (c) is completed.
</pre>


d)  &nbsp;&nbsp;What is the maximum height or the plane?


<pre>
    The highest point of the flight is the vertex of the quadratic function h(t).


    The value of "t"  is  t = " {{{-b/(2a)}}} " = {{{-18/(2*(-5))}}} = {{{-18/(-10)}}} = 1.8.


    It means that the airplane reaches the maximum height at t= 1.8 seconds.


    The maximum height is the value of the height function at t= 1.8

        {{{h[max]}}} = {{{-5*1.8^2 + 18*1.8 + 1}}} = 17.2 meters.     <U>ANSWER</U>
</pre>

The problem is just solved.  &nbsp;&nbsp;&nbsp;&nbsp;//  &nbsp;&nbsp;&nbsp;&nbsp;All questions are answered.


==================


To see many other solved similar and different problems, &nbsp;see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Introductory-lesson-on-a-projectile-thrown-shot-launched-vertically-up.lesson>Introductory lesson on a projectile thrown-shot-launched vertically up</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Problem-on-a-projectile-moving-vertically.lesson>Problem on a projectile moving vertically up and down</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Problem-on-projectile-shooted-vertically-upward.lesson>Problem on an arrow shot vertically upward</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Typical-problems-on-an-projectile-moving-vertically-up-and-down.lesson>Problem on a ball thrown vertically up from the top of a tower</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Problem-on-a-toy-rocket-launched-vertically-up--from-the-top-of-a-platform.lesson>Problem on a toy rocket launched vertically up from a tall platform</A>


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Problems-on-a-projectlie-thrown-horizontally.lesson>Problem on a projectile thrown horizontally</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Problems-on-projectile-launched-at-the-angle-to-horizon.lesson>Problems on projectile launched at an angle to horizon</A> 


Consider these lessons as your textbook, &nbsp;handbook, &nbsp;tutorials and &nbsp;(free of charge) &nbsp;home teacher.
Read them attentively and learn how to solve this type of problems once and for all.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this textbook under the topic "<U>Projectiles launched/thrown and moving vertically up and dawn</U>".



Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.