An archer shoots an arrow from ground level towards a tower of a castle 45m high. The height of the arrow above the ground after t seconds is given by the equation h = 30t - 5t²
a) after how much time does the arrow reach the top of the tower ?
b) If the arrow doesn't hit anything, after how long will it he at ground level again?
h = 30t - 5t² = 45
5 t^2 -30 t + 45 = 0
t = 3 s
when does h = 0?
5 t^2 - 30 t = 0
t (30- 5 t) = 0
t = 6 s
a)
When the arrow reach the top of the tower h = 45 m
h = 30 t - 5 t²
45 = - 5 t² + 30 t
Subtract 45 to both sides
0 = - 5 t² + 30 t - 45
Divide both sides by - 5
0 = t² - 6 t + 9
Now you must solve equation:
t² - 6 t + 9 = 0
The solution is t = 3
b)
If the arrow doesn't hit anything the arrow will fall to the ground and the height will be zero.
So slove:
30 t - 5 t² = 0
Divide both sides by - 5 t
- 6 + t =0
Add 6 to both sides
t = 6
To find the time it takes for the arrow to reach the top of the tower, we need to find the value of t when the height of the arrow (h) is equal to the height of the tower (45m).
a) Set the equation h = 30t - 5t² equal to 45:
30t - 5t² = 45
Rearranging the equation, we get:
5t² - 30t + 45 = 0
Now we need to solve this quadratic equation for t. There are multiple ways to solve quadratic equations, such as factoring, completing the square, or using the quadratic formula. In this case, we can factor out a common factor of 5:
5(t² - 6t + 9) = 0
Now we have a trinomial that can be factored:
5(t - 3)(t - 3) = 0
So, the equation simplifies to:
(t - 3)² = 0
From this, we can see that the value of t that makes the equation equal to zero is t = 3. Therefore, it takes 3 seconds for the arrow to reach the top of the tower.
b) To find the time it takes for the arrow to reach ground level again, we need to find the value of t when the height of the arrow (h) is equal to zero.
Setting the equation h = 30t - 5t² equal to zero:
30t - 5t² = 0
We can factor out a common factor of t:
t(30 - 5t) = 0
From this equation, we can see that either t = 0 or 30 - 5t = 0.
For t = 0, the arrow is at the ground level initially, so it won't be considered as the arrow reaching ground level again.
Using the second equation, we can solve for t:
30 - 5t = 0
-5t = -30
t = 6
Therefore, it takes 6 seconds for the arrow to reach ground level again if it doesn't hit anything.