The function [tex]$D(t)$[/tex] defines a traveler's distance from home, in miles, as a function of time, in hours.
[tex]$D(t)=\left\{\begin{array}{cl}
300 t+125, & 0 \leq t<2.5 \\
875, & 2.5 \leq t \leq 3.5 \\
75 t+612.5, & 3.5\end{array}\right.$[/tex]
Which times and distances are represented by the function? Select three options.
A. The starting distance, at 0 hours, is 300 miles.
B. At 2 hours, the traveler is 725 miles from home.
C. At 2.5 hours, the traveler is still moving farther from home.
D. At 3 hours, the distance is constant, at 875 miles.
E. The total distance from home after 6 hours is [tex]$1,062.5$[/tex] miles.
Answer (2)
Evaluate the distance at 0 hours: D ( 0 ) = 125 miles.
Evaluate the distance at 2 hours: D ( 2 ) = 725 miles.
Evaluate the distance at 2.5 hours: D ( 2.5 ) = 875 miles.
Evaluate the distance at 3 hours: D ( 3 ) = 875 miles.
Evaluate the distance at 6 hours: D ( 6 ) = 1062.5 miles.
The correct statements are: At 2 hours, the traveler is 725 miles from home; At 3 hours, the distance is constant, at 875 miles; The total distance from home after 6 hours is 1 , 062.5 miles.
Explanation
Problem Analysis We are given a piecewise function D ( t ) that describes a traveler's distance from home at time t . We need to verify which of the given statements are correct.
Checking Statement 1 Statement 1: The starting distance, at 0 hours, is 300 miles. We evaluate D ( 0 ) using the first piece of the function, D ( t ) = 300 t + 125 for 0 ≤ t < 2.5 .
D ( 0 ) = 300 ( 0 ) + 125 = 125 The starting distance is 125 miles, not 300 miles. So, this statement is incorrect.
Checking Statement 2 Statement 2: At 2 hours, the traveler is 725 miles from home. We evaluate D ( 2 ) using the first piece of the function, D ( t ) = 300 t + 125 for 0 ≤ t < 2.5 .
D ( 2 ) = 300 ( 2 ) + 125 = 600 + 125 = 725 The distance at 2 hours is 725 miles. So, this statement is correct.
Checking Statement 3 Statement 3: At 2.5 hours, the traveler is still moving farther from home. At t = 2.5 , the function transitions from D ( t ) = 300 t + 125 to D ( t ) = 875 . We need to determine if the distance is still increasing at t = 2.5 .
We calculate D ( 2.5 ) using the first piece: D ( 2.5 ) = 300 ( 2.5 ) + 125 = 750 + 125 = 875 At t = 2.5 , D ( 2.5 ) = 875 miles. For 2.5 ≤ t ≤ 3.5 , D ( t ) = 875 , which means the distance is constant. Therefore, the traveler is not moving farther from home at 2.5 hours. So, this statement is incorrect.
Checking Statement 4 Statement 4: At 3 hours, the distance is constant, at 875 miles. We evaluate D ( 3 ) using the second piece of the function, D ( t ) = 875 for 2.5 ≤ t ≤ 3.5 .
D ( 3 ) = 875 The distance at 3 hours is 875 miles. So, this statement is correct.
Checking Statement 5 Statement 5: The total distance from home after 6 hours is 1 , 062.5 miles. We evaluate D ( 6 ) using the third piece of the function, D ( t ) = 75 t + 612.5 for 3.5 < t ≤ 6 .
D ( 6 ) = 75 ( 6 ) + 612.5 = 450 + 612.5 = 1062.5 The distance at 6 hours is 1062.5 miles. So, this statement is correct.
Final Answer The correct statements are:
At 2 hours, the traveler is 725 miles from home.
At 3 hours, the distance is constant, at 875 miles.
The total distance from home after 6 hours is 1 , 062.5 miles.
Examples
Understanding piecewise functions is crucial in many real-world scenarios. For instance, consider a cell phone plan where you pay a fixed amount for a certain amount of data, and then the price per gigabyte increases after you exceed that limit. Similarly, delivery services often have different pricing structures based on distance or weight, which can be modeled using piecewise functions. These functions help in accurately representing costs, distances, or any quantity that changes based on specific conditions or intervals.
The correct statements are that at 2 hours, the traveler is 725 miles from home; at 3 hours, the distance is constant at 875 miles; and after 6 hours, the total distance from home is 1,062.5 miles. Statements B, D, and E are correct while A and C are not.
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