A rocket is launched from a tower. The height of the rocket, y in feet, is related to the time after launch, x in seconds, by the given equation. Using this equation, find out the time at which the rocket will reach its max, to the nearest 100th of a second.

Question

y, equals, minus, 16, x, squared, plus, 107, x, plus, 60
y=−16x
2
+107x+60
Answer
Attempt 1 out of 2

Answer 1

To find the time at which the rocket reaches its maximum height, we need to determine the vertex of the quadratic function given by the equation:

\[ y = -16x^2 + 107x + 60 \]

For a quadratic equation of the form \( y = ax^2 + bx + c \), the \( x \)-coordinate of the vertex (which gives the maximum height when \( a < 0 \)) can be calculated using the formula:

\[ x = -\frac{b}{2a} \]

In this case, the coefficients are:

• \( a = -16 \)
• \( b = 107 \)

Now plug these values into the formula:

\[ x = -\frac{107}{2 \times -16} \]

\[ x = -\frac{107}{-32} \]

\[ x = \frac{107}{32} \]

Calculating \( \frac{107}{32} \):

\[ x \approx 3.34375 \]

Now, rounding to the nearest hundredth of a second:

\[ x \approx 3.34 \text{ seconds} \]

Therefore, the time at which the rocket will reach its maximum height is approximately 3.34 seconds.