Question
An equation that models the height of an object dropped from the top of a building is y equals negative 16 x squared plus 30y=−16x2+30 where x is time in sec. Another equation, yequals=14, models the path of a bird flying in the air. Write a system of equations, and then solve to find how many seconds the object is in the air before it crosses the bird's path.
Question content area bottom
Part 1
Complete the system of equations for the object crossing the bird's path.
yequals=negative 16 x squared plus 30−16x2+30
yequals=1414
Part 2
Solve the system of the equations. The object will cross the bird's path in enter your response here second(s).
(Type an integer or decimal rounded to the nearest hundredth as needed.)
Question content area bottom
Part 1
Complete the system of equations for the object crossing the bird's path.
yequals=negative 16 x squared plus 30−16x2+30
yequals=1414
Part 2
Solve the system of the equations. The object will cross the bird's path in enter your response here second(s).
(Type an integer or decimal rounded to the nearest hundredth as needed.)
Answers
GPT-4o mini
To find when the object dropped from the top of the building crosses the path of the bird, we need to set up a system of equations based on the models provided.
### Part 1: Write the system of equations
Given:
1. The height of the object dropped from the building:
\[
y = -16x^2 + 30
\]
2. The height of the bird flying in the air:
\[
y = 14
\]
Therefore, the system of equations is:
\[
\begin{cases}
y = -16x^2 + 30 \\
y = 14
\end{cases}
\]
### Part 2: Solve the system of equations
To find out when the object crosses the bird's path, we will set both equations for \(y\) equal to each other:
\[
-16x^2 + 30 = 14
\]
Now, solve for \(x\):
1. Subtract 14 from both sides:
\[
-16x^2 + 30 - 14 = 0
\]
\[
-16x^2 + 16 = 0
\]
2. Factor out \(-16\):
\[
-16(x^2 - 1) = 0
\]
So, set \(x^2 - 1 = 0\).
3. Solve for \(x^2\):
\[
x^2 = 1
\]
Thus, taking the square root gives:
\[
x = 1 \quad \text{or} \quad x = -1
\]
Since time cannot be negative in this context, we select:
\[
x = 1
\]
### Conclusion
The object will cross the bird's path in **1 second**.
### Part 1: Write the system of equations
Given:
1. The height of the object dropped from the building:
\[
y = -16x^2 + 30
\]
2. The height of the bird flying in the air:
\[
y = 14
\]
Therefore, the system of equations is:
\[
\begin{cases}
y = -16x^2 + 30 \\
y = 14
\end{cases}
\]
### Part 2: Solve the system of equations
To find out when the object crosses the bird's path, we will set both equations for \(y\) equal to each other:
\[
-16x^2 + 30 = 14
\]
Now, solve for \(x\):
1. Subtract 14 from both sides:
\[
-16x^2 + 30 - 14 = 0
\]
\[
-16x^2 + 16 = 0
\]
2. Factor out \(-16\):
\[
-16(x^2 - 1) = 0
\]
So, set \(x^2 - 1 = 0\).
3. Solve for \(x^2\):
\[
x^2 = 1
\]
Thus, taking the square root gives:
\[
x = 1 \quad \text{or} \quad x = -1
\]
Since time cannot be negative in this context, we select:
\[
x = 1
\]
### Conclusion
The object will cross the bird's path in **1 second**.