to graph the function using the desmos.com graphing software, let y = -4.9x^2 + 166x + 2.2.
y is the height.
x is the elapsed seconds.
the graph, with all the necessary intersection points, is shown below:
you can see from the graph that the height is 2.2 meters at 0 seconds.
the graph also shows you that the maximum height is attained 16.939 seconds after firing the flare, and the maximum height attained is 1408.118 meters.
the function is H(t)=-4.9t^2+166t+2.2.
to find the height 3 seconds after launch, set t= 3 and solve for H(t).
you will get H(3) = -4.9 * 3^2 + 166 * 3 + 2.2 = 456.1 meters.
this is the height of the flare 3 seconds after firing.
with your graph, set x = 3 and the intersection of that vertical line with the graph will show the height.
the coordinate point of the intersection is (3,456.1)
that tells you the value of x is 3 and the value of y is 456.1.
x is the number of seconds after firing.
y is the height in meters.
Evaluate H(3), and explain what it means in the context of the situation
based on the height model, is it likely the sailor was sitting or standing in the raft when he fired the flare gun? How do you know?
H(3) was explained above.
it's the height of the flare 3 seconds after lanuch.
H(0) is the height of the flare at launch.
that height is 2.2 meters.
2.2 meters is equal to 7.21785 feet.
since the normal height of an individual is between 5 feet and 6 feet, then it is most likely that the sailor was standing when he shot the flare, while his arm was extended upward.
give the height from which the flare was fired in meters?
the flare was fired from a height of 2.2 meters.
give the maximum height reached by the flare? (Round to the nearest tenth)
How long did it take for the flare to reach the maximum height? (Round to the nearest tenth)
the flare reached a maximum height of 1408.1 meters 16.9 seconds after firing.
how long was the flare in the air? (Round to the nearest tenth)
the flare was in the air for 16.9 seconds from the time it was fired to the time it landed in the water, assuming that 0 meters is sea level.
A second boat was several mies away from the lost sailors, and, if the occupants of the second boat happened to be looking at the right moment in the right direction, the flare would be visible as long as it was at least 400 meters in the air. For how long would the flare have been visible to the occupants of the second boat. (Round to the nearest tenth)
from the graph, set y = 400 and you can see that the flare first attained 400 meters 2.595 second after firing and then dropped back down to 400 meters 31.282 seconds after firing.
that means it was at or above 400 meters in height 31.282 - 2.595 = 26.687 seconds.
round this to 1 decimal digit and it must have been visible for 26.7 seconds.
being able to see this from the graph is pretty easy, once the graph has been set up properly.
doing it algebraically is a little more difficult because it requires some arithmetic and some formulas to be applied.
to make the use of the applicable formulas easiwer, translate the equation from H(t) = -4.9 * t^2 + 166 * t + 2.2 to y = -4.9 * x^2 + 166 * x + 2.2.
you do his by letting y = H(t), and replacing t with x.
y is the height after x seconds.
x is the number of seconds after firing.
to find out the height of the flare at firing, set x = 0 and you will get y 2.2.
to find out when the height of the flare is 0, set y = 0 and solve the quadratic equation using the quadratic formula of x = (-b plus or minus sqrt(b^2-4ac))/(2a)
when y = 0, the equation is in standard form of ax^2 + bx + c = 0.
your equation, being -4.9x^2 + 166x + 2.2 = 0, gets you:
a = -4.9
b = 166
c = 2.2
you would factor this equation using the quadratic formula to get:
x = 33.890798851886 or x = -0.013247831477768
the value of y is equal to 0 when x equal these values.
x = minus ..... isn't possible, so you get the value of y = 0 when x = 33.891 seconds rounded to 3 decimal digits.
that's the same value the graph showed you, but now you did it by formula rather than by graphing.
the maximum height of the quadratic equation, when it is in standard form, is when x = -b/2a.
x = -b/2a becomes x = -166 / (2 * -4.9) which makes x equal to 16.93877551 seconds after launch.
the value of y when x = 16.93877551 is y = 1408.118367 meters.
that agrees with the graph value, when rounded to 3 decimal digits.
to find the value of x when the height is 400 meters, set y = 400 and you get 400 = -4.9x^2 + 166x + 2.2
put this in standard form by subtracting 400 from both sides of the equation to get:
-4.9x^2 + 166x + 2.2 - 400 = 0
simplify this to get -4.9x^2 + 166x - 397.8 = 0.
since this is now in standard form:
a = -4.9
b = 166
c = -397.8
solve this using the quadratic formula to get:
x = 2.5951901036294 or x = 31.282360916779 seconds.
this agrees with the graph values of 2.595 and 31.282.
i didn't do the calculations using the quadratic formula manually.
i cheated by using an online quadratic equation solver.
you, however, will probably need to apply the formula manually, with help from a calculator to do the arithmetic.