Math Problem Statement
Two cars are approaching an intersection. One is
11
milemile
south of the intersection and is moving at a constant speed of
4040
miles per hour. At the same time, the other car is
22
milesmiles
east of the intersection and is moving at a constant speed of
1010
miles per hour.
Question content area bottom
Part 1
(a) Express the distance d between the cars as a function of time t. (Hint: At
t equals 0 commat=0,
the cars are
11
milemile
south and
22
milesmiles
east of the intersection, respectively.)
d
equals=
StartRoot nothing t squared minus nothing t plus nothing EndRoot enter your response heret2− enter your response heret+ enter your response here
Solution
To express the distance between the two cars as a function of time , let's start by defining the positions of the two cars.
Car 1 (South of the intersection):
- At , Car 1 is 11 miles south of the intersection.
- The car is moving north at 40 miles per hour.
- So, the position of Car 1 as a function of time is: Here, represents the distance of Car 1 from the intersection (positive means it is still south of the intersection).
Car 2 (East of the intersection):
- At , Car 2 is 2 miles east of the intersection.
- The car is moving west at 10 miles per hour.
- So, the position of Car 2 as a function of time is: Here, represents the distance of Car 2 from the intersection (positive means it is still east of the intersection).
Distance between the two cars:
The distance between the two cars at time is given by the Pythagorean theorem: Substituting the expressions for and :
Simplifying the expression:
- Expand the squares: = \sqrt{(4 - 40t + 100t^2) + (121 - 880t + 1600t^2)}$$
- Combine like terms:
Therefore, the distance between the cars as a function of time is:
Would you like more details or have any questions about this?
Related Questions:
- How does the speed of the cars affect the rate at which the distance between them changes?
- At what time will the two cars be closest to each other?
- What is the minimum distance between the two cars?
- How would the function change if the cars were moving in the opposite directions?
- How does the expression change if the second car starts at a different distance from the intersection?
Tip: When solving problems involving motion, always break down the motion into components and consider each dimension separately before combining the results.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Pythagorean Theorem
Functions of Time
Formulas
Distance formula: d = √(x^2 + y^2)
Position of car 1: y_1(t) = 11 - 40t
Position of car 2: x_2(t) = 2 - 10t
Theorems
Pythagorean Theorem
Suitable Grade Level
Grades 10-12
Related Recommendation
Distance Between Two Cars Approaching an Intersection
Distance Between Two Cars Using the Law of Cosines
Rate of Change in Distance Between Two Cars Using Related Rates
Calculate Rate of Change Between a Police Car and Speeding Car
Solving Relative Motion: Two Cars Starting at Different Times and Speeds