a toy cannon ball is launched from a cannon on top of a platform . The equation h(t)=-5t^2+20t+4 gives the height h, in meters, of the bell t seconds after it is launched. what equation can be used to tell whether the ball reaches a height of 12 m?
To determine whether the ball reaches a height of 12 meters, we need to set up the equation and solve for t.
Given that h(t) = -5t^2 + 20t + 4, we can set it equal to 12:
-5t^2 + 20t + 4 = 12
Rearranging the equation:
-5t^2 + 20t + 4 - 12 = 0
-5t^2 + 20t - 8 = 0
Now we have a quadratic equation. To solve for t, we can either factor it or use the quadratic formula. To factor it, we can start by factoring out a common factor of -1:
-t^2 + 4t + 8 = 0
Now, we can try factoring by grouping. Splitting the middle term:
(-t^2 + 8t) + (4t + 8) = 0
Factoring out a common factor in each group:
-t(t - 8) + 4(t + 2) = 0
Now we can factor out the common factors:
-t(t - 8) + 4(t + 2) = 0
-t(t - 8) - 4(t - 2) = 0
Now, we have two factors:
t - 8 = 0 or t + 2 = 0
Solving for t:
t = 8 or t = -2
Therefore, the ball reaches a height of 12m either at 8 seconds or -2 seconds. However, since time cannot be negative, the ball reaches a height of 12m at t = 8 seconds.
To determine the equation that can be used to tell whether the ball reaches a height of 12 meters, we need to set up the given equation, h(t) = -5t^2 + 20t + 4, equal to the desired height, which is 12 meters.
So, we can write the equation as follows:
-5t^2 + 20t + 4 = 12
Now, we need to solve this equation to find the value(s) of t for which the ball reaches a height of 12 meters.
Rearranging the equation, we have:
-5t^2 + 20t + 4 - 12 = 0
-5t^2 + 20t - 8 = 0
Next, we can divide the entire equation by -1 to simplify it:
5t^2 - 20t + 8 = 0
Now, we can use either factoring, completing the square, or the quadratic formula to solve this equation.
Using factoring:
5t^2 - 20t + 8 = 0
(5t - 2)(t - 4) = 0
Setting each factor equal to zero:
5t - 2 = 0 or t - 4 = 0
Solving each equation:
5t = 2 or t = 4
t = 2/5 or t = 4
Therefore, the equation that can be used to determine whether the ball reaches a height of 12 meters is t = 2/5 or t = 4.
To determine whether the toy cannon ball reaches a height of 12 meters, we can set up the equation h(t) = 12 and solve for t.
Given that h(t) = -5t^2 + 20t + 4, we can substitute h(t) with 12:
12 = -5t^2 + 20t + 4
Now, rearrange the equation to solve for t:
-5t^2 + 20t + 4 - 12 = 0
Combine like terms:
-5t^2 + 20t - 8 = 0
This is a quadratic equation in standard form. To solve for t, you can use factoring, completing the square, or the quadratic formula.