Question
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 the maximum height reached by the rocket, to the nearest tenth of a foot.
y, equals, minus, 16, x, squared, plus, 162, x, plus, 82
y=−16x
2
+162x+82
Answer
Attempt 1 out of 5
y, equals, minus, 16, x, squared, plus, 162, x, plus, 82
y=−16x
2
+162x+82
Answer
Attempt 1 out of 5
Answers
GPT-4o mini
To find the maximum height reached by the rocket, we can analyze the quadratic equation given:
\[ y = -16x^2 + 162x + 82 \]
This equation represents a parabola that opens downwards (since the coefficient of \(x^2\) is negative). The maximum height can be found at the vertex of this parabola.
The x-coordinate of the vertex of a quadratic equation in the form \(ax^2 + bx + c\) can be calculated using the formula:
\[ x = -\frac{b}{2a} \]
In this case, \(a = -16\) and \(b = 162\).
Plugging in the values:
\[
x = -\frac{162}{2 \cdot -16} = \frac{162}{32} = 5.0625
\]
Now, we need to substitute this value of \(x\) back into the original equation to find the maximum height \(y\):
\[
y = -16(5.0625)^2 + 162(5.0625) + 82
\]
Calculating \( (5.0625)^2 \):
\[
(5.0625)^2 = 25.62890625
\]
Now substituting that back into the equation:
\[
y = -16(25.62890625) + 162(5.0625) + 82
\]
Calculating each term:
1. \( -16 \times 25.62890625 = -409.0225 \)
2. \( 162 \times 5.0625 = 819.375 \)
3. Adding these results to 82:
\[
y = -409.0225 + 819.375 + 82
\]
Calculating \(y\):
\[
y = -409.0225 + 901.375 = 492.3525
\]
Rounding to the nearest tenth of a foot, the maximum height reached by the rocket is:
\[
\boxed{492.4 \text{ feet}}
\]
\[ y = -16x^2 + 162x + 82 \]
This equation represents a parabola that opens downwards (since the coefficient of \(x^2\) is negative). The maximum height can be found at the vertex of this parabola.
The x-coordinate of the vertex of a quadratic equation in the form \(ax^2 + bx + c\) can be calculated using the formula:
\[ x = -\frac{b}{2a} \]
In this case, \(a = -16\) and \(b = 162\).
Plugging in the values:
\[
x = -\frac{162}{2 \cdot -16} = \frac{162}{32} = 5.0625
\]
Now, we need to substitute this value of \(x\) back into the original equation to find the maximum height \(y\):
\[
y = -16(5.0625)^2 + 162(5.0625) + 82
\]
Calculating \( (5.0625)^2 \):
\[
(5.0625)^2 = 25.62890625
\]
Now substituting that back into the equation:
\[
y = -16(25.62890625) + 162(5.0625) + 82
\]
Calculating each term:
1. \( -16 \times 25.62890625 = -409.0225 \)
2. \( 162 \times 5.0625 = 819.375 \)
3. Adding these results to 82:
\[
y = -409.0225 + 819.375 + 82
\]
Calculating \(y\):
\[
y = -409.0225 + 901.375 = 492.3525
\]
Rounding to the nearest tenth of a foot, the maximum height reached by the rocket is:
\[
\boxed{492.4 \text{ feet}}
\]