A firework is launched from a cliff overlooking the ocean. The elevation of the firework shell, in feet above sea level, t seconds after launch is given by the expression -16t^2+157t+63. How high is the top of the cliff, in feet above sea level.
We just started and my teacher wasn't here today.
Who needs a teacher? Just think a bit. Where is the ball to start with? At the top of the cliff.
That's when t=0.
So, plug in 0 for t and evaluate.
63
To find the height of the top of the cliff, we need to determine the highest point of the firework's trajectory. This is also known as the vertex of the quadratic function.
The given expression represents the height of the firework shell, in feet above sea level, as a function of time (t), which is a quadratic function. The general form of a quadratic function is given by:
f(t) = at^2 + bt + c
In this case, the expression -16t^2 + 157t + 63 is already in the form f(t) = at^2 + bt + c, where a = -16, b = 157, and c = 63.
To find the vertex of the quadratic function, we can use the formula:
t = -b / 2a
Plugging in the values, we have:
t = -157 / (2 * -16)
Simplifying further:
t = -157 / -32
t = 4.90625
Now, to find the height of the top of the cliff, we substitute this value of t into the given expression:
f(4.90625) = -16(4.90625)^2 + 157(4.90625) + 63
Calculating further:
f(4.90625) = -16(24.07) + 770.9375 + 63
f(4.90625) = -385.12 + 770.9375 + 63
f(4.90625) = 448.8175
Therefore, the height of the top of the cliff, in feet above sea level, is approximately 448.82 feet.
To find the height of the top of the cliff, we need to determine the maximum point of the function that represents the elevation of the firework shell.
The general form of a quadratic function is given by f(t) = at^2 + bt + c, where "a" represents the coefficient of the t^2 term, "b" represents the coefficient of the t term, and "c" represents the constant term.
In this case, the function representing the elevation of the firework shell is f(t) = -16t^2 + 157t + 63.
To find the maximum point of the function, we can use the vertex formula. The x-coordinate of the vertex can be determined by the formula x = -b / (2a).
In our case, "a" is -16 and "b" is 157. Substituting the values into the formula, we have x = -(157) / (2(-16)) = 4.90625.
Now we need to find the y-coordinate of the vertex, which represents the maximum height of the firework shell. To do this, we substitute the x-coordinate value back into the function: f(4.90625) = -16(4.90625)^2 + 157(4.90625) + 63.
Evaluating this expression will give us the maximum height of the firework shell, which is also the height of the top of the cliff.