Problem 8
Question
truck is 300 miles due east of a car and is traveling west at the constant speed of 30 miles per hour. Meanwhile, the car is going north at the constant speed of 60 miles per hour. Express the distance between the car and truck as a function of time.
Step-by-Step Solution
Verified Answer
\[ d(t) = \sqrt{4500t^2 - 18000t + 90000} \]
1Step 1 - Define Variables
Let t represent the time in hours. Define the position of the truck and the car with respect to time.
2Step 2 - Position of the Truck
The initial position of the truck is 300 miles east. Since it is traveling west at 30 miles per hour, its position as a function of time is given by \[ x_{truck}(t) = 300 - 30t \]
3Step 3 - Position of the Car
The car starts from the origin and is moving north at 60 miles per hour. Therefore, its position as a function of time is given by \[ y_{car}(t) = 60t \]
4Step 4 - Calculate the Distance
To find the distance between the car and the truck, use the distance formula: \[ d(t) = \sqrt{(x_{truck}(t) - 0)^2 + (0 - y_{car}(t))^2} \]. Substituting the positions, \[ d(t) = \sqrt{(300 - 30t)^2 + (60t)^2} \]
5Step 5 - Simplify the Expression
Simplify the expression to get the final function: \[ d(t) = \sqrt{(300 - 30t)^2 + (60t)^2} = \sqrt{90000 - 18000t + 900t^2 + 3600t^2} = \sqrt{4500t^2 - 18000t + 90000} \]. This is the distance function.
Key Concepts
Position FunctionsDistance FormulaSimplification of Expressions
Position Functions
To determine the distance between two moving objects, we first need to define how their positions change over time. A position function helps us describe the location of an object at any given time.
In this exercise, the truck starts 300 miles due east of the car and moves westward. This travel direction and constant speed allow us to set up its position function as:
\[ x_{truck}(t) = 300 - 30t \]
Here, 300 is the truck's starting position, and 30 is the speed at which it moves west, thus the minus sign indicating it is traveling in the negative direction of the x-axis.
In contrast, the car moves north from the origin. Since it starts from zero and moves northward at a speed of 60 miles per hour, its position function is:
\[ y_{car}(t) = 60t \]
Understanding how to establish position functions is crucial, as they lay the foundation for calculating changing distances between moving objects.
In this exercise, the truck starts 300 miles due east of the car and moves westward. This travel direction and constant speed allow us to set up its position function as:
\[ x_{truck}(t) = 300 - 30t \]
Here, 300 is the truck's starting position, and 30 is the speed at which it moves west, thus the minus sign indicating it is traveling in the negative direction of the x-axis.
In contrast, the car moves north from the origin. Since it starts from zero and moves northward at a speed of 60 miles per hour, its position function is:
\[ y_{car}(t) = 60t \]
Understanding how to establish position functions is crucial, as they lay the foundation for calculating changing distances between moving objects.
Distance Formula
Once position functions are defined, we can use the distance formula to calculate the distance between the two objects. The distance formula in two dimensions is derived from the Pythagorean theorem:
\[ d(t) = \sqrt{(x_{2}(t) - x_{1}(t))^2 + (y_{2}(t) - y_{1}(t))^2} \]
In this case, the truck and car are moving perpendicularly. Their distance at any time t is calculated as:
\[ d(t) = \sqrt{(x_{truck}(t) - 0)^2 + (0 - y_{car}(t))^2} \]
Substituting the position functions we already discovered:
\[ d(t) = \sqrt{(300 - 30t)^2 + (60t)^2} \]
This formula gives a mathematical way to determine the exact separation between the truck and car at every instance of time.
Understanding the distance formula helps students see how seemingly complex movements can be broken down into simple equations that consider both horizontal and vertical components of distance.
\[ d(t) = \sqrt{(x_{2}(t) - x_{1}(t))^2 + (y_{2}(t) - y_{1}(t))^2} \]
In this case, the truck and car are moving perpendicularly. Their distance at any time t is calculated as:
\[ d(t) = \sqrt{(x_{truck}(t) - 0)^2 + (0 - y_{car}(t))^2} \]
Substituting the position functions we already discovered:
\[ d(t) = \sqrt{(300 - 30t)^2 + (60t)^2} \]
This formula gives a mathematical way to determine the exact separation between the truck and car at every instance of time.
Understanding the distance formula helps students see how seemingly complex movements can be broken down into simple equations that consider both horizontal and vertical components of distance.
Simplification of Expressions
After substituting the position functions into the distance formula, we end up with the formula that looks quite complex. Our job is to simplify this expression to make it more understandable and usable.
Let's simplify:
First, expand each term inside the square root:
\[ d(t) = \sqrt{(300 - 30t)^2 + (60t)^2} \]
After expanding, we get:
\[ (300 - 30t)^2 = 90000 - 18000t + 900t^2 \]
\[ (60t)^2 = 3600t^2 \]
Combining these results into one equation under the square root:
\[ d(t) = \sqrt{900t^2 - 18000t + 90000 + 3600t^2} \]
Combine like terms:
\[ d(t) = \sqrt{4500t^2 - 18000t + 90000} \]
Now we’ve simplified the expression into a more manageable form:
Simplification of expressions allows us to see clearer relationships between variables and makes further calculations easier. It is important for both solving equations and understanding underlying trends in the data.
Let's simplify:
First, expand each term inside the square root:
\[ d(t) = \sqrt{(300 - 30t)^2 + (60t)^2} \]
After expanding, we get:
\[ (300 - 30t)^2 = 90000 - 18000t + 900t^2 \]
\[ (60t)^2 = 3600t^2 \]
Combining these results into one equation under the square root:
\[ d(t) = \sqrt{900t^2 - 18000t + 90000 + 3600t^2} \]
Combine like terms:
\[ d(t) = \sqrt{4500t^2 - 18000t + 90000} \]
Now we’ve simplified the expression into a more manageable form:
Simplification of expressions allows us to see clearer relationships between variables and makes further calculations easier. It is important for both solving equations and understanding underlying trends in the data.
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