Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Get immediate and reliable solutions to your questions from a community of experienced experts on our Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To find the average velocity of an object moving along a line given by the function [tex]\( s(t) = -172 + 102t \)[/tex] over various intervals, follow these steps:
### Interval [1, 8]
First, calculate the average velocity over the interval [1, 8].
Step 1: Identify the position function [tex]\( s(t) \)[/tex].
[tex]\[ s(t) = -172 + 102t \][/tex]
Step 2: Evaluate the position at the beginning and end of the time interval.
- For [tex]\( t = 1 \)[/tex]:
[tex]\[ s(1) = -172 + 102 \cdot 1 = -172 + 102 = -70 \][/tex]
- For [tex]\( t = 8 \)[/tex]:
[tex]\[ s(8) = -172 + 102 \cdot 8 = -172 + 816 = 644 \][/tex]
Step 3: Compute the change in position [tex]\(\Delta s\)[/tex].
[tex]\[ \Delta s = s(8) - s(1) = 644 - (-70) = 644 + 70 = 714 \][/tex]
Step 4: Compute the change in time [tex]\(\Delta t\)[/tex].
[tex]\[ \Delta t = 8 - 1 = 7 \][/tex]
Step 5: Calculate the average velocity [tex]\(\bar{v}\)[/tex].
[tex]\[ \bar{v} = \frac{\Delta s}{\Delta t} = \frac{714}{7} = 102 \][/tex]
Therefore, the average velocity of the object over the interval [tex]\([1, 8]\)[/tex] is [tex]\( 102 \, \text{units of distance per unit time} \)[/tex].
### Interval [1, 7]
Repeat the steps for the interval [1, 7]:
Step 1: Evaluate the position at [tex]\( t = 1 \)[/tex] and [tex]\( t = 7 \)[/tex].
- For [tex]\( t = 1 \)[/tex]:
[tex]\[ s(1) = -172 + 102 = -70 \][/tex]
- For [tex]\( t = 7 \)[/tex]:
[tex]\[ s(7) = -172 + 714 = 542 \][/tex]
Step 2: Compute [tex]\(\Delta s\)[/tex].
[tex]\[ \Delta s = s(7) - s(1) = 542 - (-70) = 542 + 70 = 612 \][/tex]
Step 3: Compute [tex]\(\Delta t\)[/tex].
[tex]\[ \Delta t = 7 - 1 = 6 \][/tex]
Step 4: Calculate the average velocity.
[tex]\[ \bar{v} = \frac{\Delta s}{\Delta t} = \frac{612}{6} = 102 \][/tex]
Therefore, the average velocity over the interval [tex]\([1, 7]\)[/tex] is [tex]\( 102 \, \text{units of distance per unit time} \)[/tex].
### Interval [1, 6]
Repeat the process for the interval [1, 6]:
Step 1: Evaluate the position at [tex]\( t = 1 \)[/tex] and [tex]\( t = 6 \)[/tex].
- For [tex]\( t = 1 \)[/tex]:
[tex]\[ s(1) = -70 \][/tex]
- For [tex]\( t = 6 \)[/tex]:
[tex]\[ s(6) = -172 + 612 = 440 \][/tex]
Step 2: Compute [tex]\(\Delta s\)[/tex].
[tex]\[ \Delta s = s(6) - s(1) = 440 - (-70) = 440 + 70 = 510 \][/tex]
Step 3: Compute [tex]\(\Delta t\)[/tex].
[tex]\[ \Delta t = 6 - 1 = 5 \][/tex]
Step 4: Calculate the average velocity.
[tex]\[ \bar{v} = \frac{\Delta s}{\Delta t} = \frac{510}{5} = 102 \][/tex]
Therefore, the average velocity over the interval [tex]\([1, 6]\)[/tex] is [tex]\( 102 \, \text{units of distance per unit time} \)[/tex].
### Interval [1, 1+h], where [tex]\( h > 0 \)[/tex]
For the general interval [tex]\([1, 1 + h]\)[/tex]:
Step 1: Evaluate the position at [tex]\( t = 1 \)[/tex] and [tex]\( t = 1 + h \)[/tex].
- For [tex]\( t = 1 \)[/tex]:
[tex]\[ s(1) = -70 \][/tex]
- For [tex]\( t = 1 + h \)[/tex]:
[tex]\[ s(1 + h) = -172 + 102(1 + h) = -172 + 102 + 102h = -70 + 102h \][/tex]
Step 2: Compute [tex]\(\Delta s\)[/tex].
[tex]\[ \Delta s = s(1 + h) - s(1) = (-70 + 102h) - (-70) = 102h \][/tex]
Step 3: Compute [tex]\(\Delta t\)[/tex].
[tex]\[ \Delta t = (1 + h) - 1 = h \][/tex]
Step 4: Calculate the average velocity.
[tex]\[ \bar{v} = \frac{\Delta s}{\Delta t} = \frac{102h}{h} = 102 \][/tex]
Therefore, the average velocity over the interval [tex]\([1, 1 + h]\)[/tex] is [tex]\( 102 \, \text{units of distance per unit time} \)[/tex].
In summary, for all given intervals, the average velocity of the object is [tex]\( 102 \, \text{units of distance per unit time} \)[/tex].
### Interval [1, 8]
First, calculate the average velocity over the interval [1, 8].
Step 1: Identify the position function [tex]\( s(t) \)[/tex].
[tex]\[ s(t) = -172 + 102t \][/tex]
Step 2: Evaluate the position at the beginning and end of the time interval.
- For [tex]\( t = 1 \)[/tex]:
[tex]\[ s(1) = -172 + 102 \cdot 1 = -172 + 102 = -70 \][/tex]
- For [tex]\( t = 8 \)[/tex]:
[tex]\[ s(8) = -172 + 102 \cdot 8 = -172 + 816 = 644 \][/tex]
Step 3: Compute the change in position [tex]\(\Delta s\)[/tex].
[tex]\[ \Delta s = s(8) - s(1) = 644 - (-70) = 644 + 70 = 714 \][/tex]
Step 4: Compute the change in time [tex]\(\Delta t\)[/tex].
[tex]\[ \Delta t = 8 - 1 = 7 \][/tex]
Step 5: Calculate the average velocity [tex]\(\bar{v}\)[/tex].
[tex]\[ \bar{v} = \frac{\Delta s}{\Delta t} = \frac{714}{7} = 102 \][/tex]
Therefore, the average velocity of the object over the interval [tex]\([1, 8]\)[/tex] is [tex]\( 102 \, \text{units of distance per unit time} \)[/tex].
### Interval [1, 7]
Repeat the steps for the interval [1, 7]:
Step 1: Evaluate the position at [tex]\( t = 1 \)[/tex] and [tex]\( t = 7 \)[/tex].
- For [tex]\( t = 1 \)[/tex]:
[tex]\[ s(1) = -172 + 102 = -70 \][/tex]
- For [tex]\( t = 7 \)[/tex]:
[tex]\[ s(7) = -172 + 714 = 542 \][/tex]
Step 2: Compute [tex]\(\Delta s\)[/tex].
[tex]\[ \Delta s = s(7) - s(1) = 542 - (-70) = 542 + 70 = 612 \][/tex]
Step 3: Compute [tex]\(\Delta t\)[/tex].
[tex]\[ \Delta t = 7 - 1 = 6 \][/tex]
Step 4: Calculate the average velocity.
[tex]\[ \bar{v} = \frac{\Delta s}{\Delta t} = \frac{612}{6} = 102 \][/tex]
Therefore, the average velocity over the interval [tex]\([1, 7]\)[/tex] is [tex]\( 102 \, \text{units of distance per unit time} \)[/tex].
### Interval [1, 6]
Repeat the process for the interval [1, 6]:
Step 1: Evaluate the position at [tex]\( t = 1 \)[/tex] and [tex]\( t = 6 \)[/tex].
- For [tex]\( t = 1 \)[/tex]:
[tex]\[ s(1) = -70 \][/tex]
- For [tex]\( t = 6 \)[/tex]:
[tex]\[ s(6) = -172 + 612 = 440 \][/tex]
Step 2: Compute [tex]\(\Delta s\)[/tex].
[tex]\[ \Delta s = s(6) - s(1) = 440 - (-70) = 440 + 70 = 510 \][/tex]
Step 3: Compute [tex]\(\Delta t\)[/tex].
[tex]\[ \Delta t = 6 - 1 = 5 \][/tex]
Step 4: Calculate the average velocity.
[tex]\[ \bar{v} = \frac{\Delta s}{\Delta t} = \frac{510}{5} = 102 \][/tex]
Therefore, the average velocity over the interval [tex]\([1, 6]\)[/tex] is [tex]\( 102 \, \text{units of distance per unit time} \)[/tex].
### Interval [1, 1+h], where [tex]\( h > 0 \)[/tex]
For the general interval [tex]\([1, 1 + h]\)[/tex]:
Step 1: Evaluate the position at [tex]\( t = 1 \)[/tex] and [tex]\( t = 1 + h \)[/tex].
- For [tex]\( t = 1 \)[/tex]:
[tex]\[ s(1) = -70 \][/tex]
- For [tex]\( t = 1 + h \)[/tex]:
[tex]\[ s(1 + h) = -172 + 102(1 + h) = -172 + 102 + 102h = -70 + 102h \][/tex]
Step 2: Compute [tex]\(\Delta s\)[/tex].
[tex]\[ \Delta s = s(1 + h) - s(1) = (-70 + 102h) - (-70) = 102h \][/tex]
Step 3: Compute [tex]\(\Delta t\)[/tex].
[tex]\[ \Delta t = (1 + h) - 1 = h \][/tex]
Step 4: Calculate the average velocity.
[tex]\[ \bar{v} = \frac{\Delta s}{\Delta t} = \frac{102h}{h} = 102 \][/tex]
Therefore, the average velocity over the interval [tex]\([1, 1 + h]\)[/tex] is [tex]\( 102 \, \text{units of distance per unit time} \)[/tex].
In summary, for all given intervals, the average velocity of the object is [tex]\( 102 \, \text{units of distance per unit time} \)[/tex].
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
