Scientific Notation Portfolio Worksheet
1. Operations with Scientific Notation:
Let's assume the following scientific numbers for use:
- \( 3.0 \times 10^3 \) (3000)
- \( 2.5 \times 10^2 \) (250)
- \( 1.2 \times 10^4 \) (12000)
- \( 4.0 \times 10^1 \) (40)
- \( 6.5 \times 10^3 \) (6500)
a. Addition: Combine \( 3.0 \times 10^3 \) and \( 6.5 \times 10^3 \).
\[ 3.0 \times 10^3 + 6.5 \times 10^3 = (3.0 + 6.5) \times 10^3 = 9.5 \times 10^3 \]
b. Subtraction: Subtract \( 2.5 \times 10^2 \) from \( 1.2 \times 10^4 \).
Convert \( 1.2 \times 10^4 \) to the same exponent as \( 2.5 \times 10^2 \):
\[ 1.2 \times 10^4 = 12000 \quad \text{and} \quad 2.5 \times 10^2 = 250 \]
Now perform the subtraction:
\[ 12000 - 250 = 11750 \]
Convert back to scientific notation:
\[ 11750 = 1.175 \times 10^4 \]
c. Multiplication: Multiply \( 1.2 \times 10^4 \) and \( 4.0 \times 10^1 \).
\[ (1.2 \times 4.0) \times (10^4 \times 10^1) = 4.8 \times 10^{4+1} = 4.8 \times 10^5 \]
d. Division: Divide \( 6.5 \times 10^3 \) by \( 4.0 \times 10^1 \).
\[ \frac{6.5}{4.0} \times 10^{3-1} = 1.625 \times 10^2 \]
e. Add a scientific number to a decimal: Add \( 3.0 \times 10^3 \) to 32.8.
Convert \( 32.8 \) to scientific notation:
\[ 32.8 = 3.28 \times 10^1 \]
To add \( 3.0 \times 10^3 \):
\[ 3.0 \times 10^3 + 3.28 \times 10^1 = 3.0 \times 10^3 + 0.0328 \times 10^3 = 3.0328 \times 10^3 \]
f. Subtract a scientific number from a decimal: Subtract \( 2.5 \times 10^2 \) from \( 1845.19 \).
First, convert \( 1845.19 \):
\[ 1845.19 = 1.84519 \times 10^3 \]
Now perform the subtraction:
\[ 1.84519 \times 10^3 - 0.250 \times 10^3 = 1.59519 \times 10^3 \]
g. Multiply a scientific number by a decimal: Multiply \( 3.0 \times 10^3 \) by \( 256.7 \).
Convert \( 256.7 \):
\[ 256.7 = 2.567 \times 10^2 \]
Multiply:
\[ (3.0 \times 2.567) \times (10^3 \times 10^2) = 7.701 \times 10^5 \]
h. Divide a scientific number by a decimal: Divide \( 1.2 \times 10^4 \) by \( 0.3 \).
Convert \( 0.3 \):
\[ 0.3 = 3.0 \times 10^{-1} \]
Now divide:
\[ \frac{1.2 \times 10^4}{3.0 \times 10^{-1}} = \frac{1.2}{3.0} \times 10^{4 - (-1)} = 0.4 \times 10^5 = 4.0 \times 10^4 \]
2. Trip Planning with Distances in Scientific Notation:
Given Distances in Scientific Notation:
| City | Distance (miles) | |--------------|-------------------| | Atlanta | \( 0 \) | | Baltimore | \( 1.2 \times 10^2 \) | | Boston | \( 1.0 \times 10^3 \) | | Chicago | \( 1.7 \times 10^3 \) | | Los Angeles | \( 3.0 \times 10^3 \) | | Seattle | \( 1.3 \times 10^4 \) | | Tampa | \( 1.9 \times 10^2 \) |
a. Trip Planning: Let’s travel from Atlanta to Chicago, then to Boston, and return to Atlanta.
Total distance traveled:
\[ \text{Distance from Atlanta to Chicago} = 1.7 \times 10^3 \] \[ \text{Distance from Chicago to Boston} = 1.0 \times 10^3 \] \[ \text{Return distance from Boston to Atlanta} = 0 \text{ (Virtually considered since we are back to the start)} \]
Total distance:
\[ \text{Total Distance} = 1.7 \times 10^3 + 1.0 \times 10^3 + 0 \Rightarrow 2.7 \times 10^3 \text{ miles} \]
In decimal notation:
\[ 2.7 \times 10^3 = 2700 \text{ miles} \]
b. Total distance for Sita's trips: Sita makes three trips from Los Angeles to Seattle.
Distance one way:
\[ 3.0 \times 10^3 = 3000 \text{ miles} \]
Total for three trips:
\[ 3.0 \times 10^3 \times 3 = 9.0 \times 10^3 \text{ miles} \]
In decimal notation:
\[ 9.0 \times 10^3 = 9000 \text{ miles} \]
c. Word Problem: If a truck can cover a distance of \( 1.5 \times 10^3 \) miles in a week, how many weeks will it take to cover \( 6.0 \times 10^3 \) miles?
Solution:
Divide total distance by distance per week:
\[ \frac{6.0 \times 10^3}{1.5 \times 10^3} = 4 \text{ weeks} \]
3. Distances of Planets from the Sun in Scientific Notation (Example Distances):
| Planet | Distance (AU) | |---------------|------------------| | Mercury | \( 5.8 \times 10^{-3} \) | | Venus | \( 1.1 \times 10^{-2} \) | | Earth | \( 1.5 \times 10^{-2} \) | | Mars | \( 2.3 \times 10^{-2} \) |
c. Subtracting distances: Subtract the distance of Mars from Venus.
\[ 1.1 \times 10^{-2} - 2.3 \times 10^{-2} = -1.2 \times 10^{-2} \]
d. Multiply distances: Multiply the distance of Earth by Mars.
\[ (1.5 \times 10^{-2}) \times (2.3 \times 10^{-2}) = 3.45 \times 10^{-4} \]
e. Divide distances: Divide the distance of Venus by Mercury.
\[ \frac{1.1 \times 10^{-2}}{5.8 \times 10^{-3}} \approx 1.898 \approx 1.9 \]
This concludes the worksheet with all the required calculations and notations.