Question
A car accelerates after being completely stopped at a stop sign and enters the highway. The distance the car has traveled in miles after x minutes is represented by y equals 0.5 x squared .y=0.5x2. A truck is traveling in the same direction at a constant speed so that its distance in miles from the same stop sign after x minutes is represented by y equals x plus 4.y=x+4. After how many minutes will the car pass the truck? Explain.
Question content area bottom
Part 1
Select the correct answer below and, if necessary, fill in the answer box within your choice.
A.
The car passes the truck in enter your response here minute(s). There is only one meaningful solution for the system of equations.
B.
The car passes the truck in enter your response here minute(s). There are two real solutions for the system of equations.
(Use a comma to separate answers as needed.)
C.
The car does not pass the truck as the system of equations does not have a real solution.
Question content area bottom
Part 1
Select the correct answer below and, if necessary, fill in the answer box within your choice.
A.
The car passes the truck in enter your response here minute(s). There is only one meaningful solution for the system of equations.
B.
The car passes the truck in enter your response here minute(s). There are two real solutions for the system of equations.
(Use a comma to separate answers as needed.)
C.
The car does not pass the truck as the system of equations does not have a real solution.
Answers
GPT-4o mini
To determine when the car passes the truck, we need to find the value of \( x \) where the distance traveled by both the car and the truck is equal.
We have the equations for both the car and the truck:
1. The distance traveled by the car:
\[
y = 0.5x^2
\]
2. The distance traveled by the truck:
\[
y = x + 4
\]
To find when the car passes the truck, we set the two equations equal to each other:
\[
0.5x^2 = x + 4
\]
Now, we will rearrange the equation to form a standard quadratic equation:
\[
0.5x^2 - x - 4 = 0
\]
To eliminate the decimal, we can multiply the entire equation by 2:
\[
x^2 - 2x - 8 = 0
\]
Next, we can factor this quadratic equation:
\[
(x - 4)(x + 2) = 0
\]
Setting each factor to zero gives us the possible solutions for \( x \):
1. \( x - 4 = 0 \) → \( x = 4 \)
2. \( x + 2 = 0 \) → \( x = -2 \)
Since time (in minutes) cannot be negative, we discard \( x = -2 \) and only consider \( x = 4 \).
Now we can conclude:
**Part 1**
A. The car passes the truck in **4 minute(s)**. There is only one meaningful solution for the system of equations.
We have the equations for both the car and the truck:
1. The distance traveled by the car:
\[
y = 0.5x^2
\]
2. The distance traveled by the truck:
\[
y = x + 4
\]
To find when the car passes the truck, we set the two equations equal to each other:
\[
0.5x^2 = x + 4
\]
Now, we will rearrange the equation to form a standard quadratic equation:
\[
0.5x^2 - x - 4 = 0
\]
To eliminate the decimal, we can multiply the entire equation by 2:
\[
x^2 - 2x - 8 = 0
\]
Next, we can factor this quadratic equation:
\[
(x - 4)(x + 2) = 0
\]
Setting each factor to zero gives us the possible solutions for \( x \):
1. \( x - 4 = 0 \) → \( x = 4 \)
2. \( x + 2 = 0 \) → \( x = -2 \)
Since time (in minutes) cannot be negative, we discard \( x = -2 \) and only consider \( x = 4 \).
Now we can conclude:
**Part 1**
A. The car passes the truck in **4 minute(s)**. There is only one meaningful solution for the system of equations.