Math Problem Statement

The position of an object in feet at time t in seconds is defined by the function s(t) = t^2 - 4t + 3. Find the average velocity of the object on the interval [2, 5]. Include units in your final answer.

Solution

To find the average velocity of the object on the interval [2,5][2, 5], we will use the formula for average velocity:

Average velocity=s(b)s(a)ba\text{Average velocity} = \frac{s(b) - s(a)}{b - a}

where s(t)s(t) is the position function, and [a,b][a, b] is the time interval. In this case, s(t)=t24t+3s(t) = t^2 - 4t + 3, and the interval is [2,5][2, 5].

Step 1: Find s(5)s(5)

s(5)=524(5)+3=2520+3=8s(5) = 5^2 - 4(5) + 3 = 25 - 20 + 3 = 8

Step 2: Find s(2)s(2)

s(2)=224(2)+3=48+3=1s(2) = 2^2 - 4(2) + 3 = 4 - 8 + 3 = -1

Step 3: Calculate the average velocity

Average velocity=s(5)s(2)52=8(1)52=8+13=93=3feet per second\text{Average velocity} = \frac{s(5) - s(2)}{5 - 2} = \frac{8 - (-1)}{5 - 2} = \frac{8 + 1}{3} = \frac{9}{3} = 3 \, \text{feet per second}

Final Answer:

The average velocity of the object on the interval [2,5][2, 5] is 3feet per second3 \, \text{feet per second}.

Would you like more details on any of these steps or have any other questions?

Here are 5 related questions:

  1. How do you find the instantaneous velocity from the position function?
  2. What is the difference between average velocity and instantaneous velocity?
  3. How would the average velocity change if the interval was [1,5][1, 5] instead of [2,5][2, 5]?
  4. What is the significance of the sign (positive or negative) of velocity?
  5. How can you interpret the position function graphically to understand velocity?

Tip: The average velocity is the slope of the secant line connecting two points on the position function graph over a given interval.

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Math Problem Analysis

Mathematical Concepts

Calculus
Average Velocity
Position Function

Formulas

Average velocity = (s(b) - s(a)) / (b - a)

Theorems

Average Rate of Change

Suitable Grade Level

Grades 9-12