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Answer :
To find the speed that maximizes the flow rate of cars on a highway, we need to analyze the given flow rate function:
[tex]\[ F = \frac{1,000v}{33 + 0.02v^2} \][/tex]
Here, [tex]\( F \)[/tex] is the flow rate of cars per hour, and [tex]\( v \)[/tex] is the speed of the traffic in miles per hour.
### Steps to solve:
1. Understand the Problem:
- We want to find the speed [tex]\( v \)[/tex] that maximizes the function [tex]\( F(v) = \frac{1,000v}{33 + 0.02v^2} \)[/tex].
2. Concept of Optimization:
- To maximize or minimize a function, we typically find its derivative, set it equal to zero, and solve for the critical points. These points are potential candidates for maximum or minimum values.
3. Find the Derivative:
- Compute the derivative of [tex]\( F \)[/tex] with respect to [tex]\( v \)[/tex], noted as [tex]\( F'(v) \)[/tex].
4. Set the Derivative Equal to Zero:
- Solve the equation [tex]\( F'(v) = 0 \)[/tex] to find the critical points of the function.
5. Evaluate Critical Points:
- Once you have the critical points, you evaluate the function [tex]\( F \)[/tex] at each critical point to determine which point gives the highest flow rate, as we are looking for a maximum.
6. Choosing the Best Speed:
- Among the critical points, identify the speed that results in the maximum flow rate.
By following this approach with careful calculations, it turns out that the speed that maximizes the flow rate of cars on this highway is 41 miles per hour.
[tex]\[ F = \frac{1,000v}{33 + 0.02v^2} \][/tex]
Here, [tex]\( F \)[/tex] is the flow rate of cars per hour, and [tex]\( v \)[/tex] is the speed of the traffic in miles per hour.
### Steps to solve:
1. Understand the Problem:
- We want to find the speed [tex]\( v \)[/tex] that maximizes the function [tex]\( F(v) = \frac{1,000v}{33 + 0.02v^2} \)[/tex].
2. Concept of Optimization:
- To maximize or minimize a function, we typically find its derivative, set it equal to zero, and solve for the critical points. These points are potential candidates for maximum or minimum values.
3. Find the Derivative:
- Compute the derivative of [tex]\( F \)[/tex] with respect to [tex]\( v \)[/tex], noted as [tex]\( F'(v) \)[/tex].
4. Set the Derivative Equal to Zero:
- Solve the equation [tex]\( F'(v) = 0 \)[/tex] to find the critical points of the function.
5. Evaluate Critical Points:
- Once you have the critical points, you evaluate the function [tex]\( F \)[/tex] at each critical point to determine which point gives the highest flow rate, as we are looking for a maximum.
6. Choosing the Best Speed:
- Among the critical points, identify the speed that results in the maximum flow rate.
By following this approach with careful calculations, it turns out that the speed that maximizes the flow rate of cars on this highway is 41 miles per hour.
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