The traffic flow rate (cars per hour) across an intersection is r(t) = 400 + 800t – 120t, where t is in hours, and t=0 is 6am. How many cars pass through the intersection between 6 am and 7 am? cars
Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
![**Problem Statement:**
The traffic flow rate (cars per hour) across an intersection is \( r(t) = 400 + 800t - 120t^2 \), where \( t \) is in hours, and \( t = 0 \) is 6 am. How many cars pass through the intersection between 6 am and 7 am?
**Diagram/Graphs Explanation:**
In this problem, there is no graph or diagram present. The problem provides a quadratic function describing the rate of traffic flow over time, and the task is to calculate the total number of cars that pass through an intersection between 6 am and 7 am.
**Solution Approach:**
To solve this problem, you integrate the given rate function \( r(t) \) from t=0 to t=1 to find the total number of cars passing through the intersection during the given time interval.
Formally, you calculate:
\[ \int_{0}^{1} (400 + 800t - 120t^2) \, dt \]
**Answer Box:**
\[ \text{\_\_\_\_\_\_\_ cars} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb53d735-0a0d-4695-97c8-e09c334fb263%2Fb37bb13d-09aa-4f9a-8c2c-f6c5a251c921%2F1xs0uzs_processed.png&w=3840&q=75)
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