Part A:
To estimate how many times greater Mount Everest's height is compared to Mount Olympus's height using scientific notation, we first need the heights in scientific notation:
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Mount Everest:
- Height = 8,849.87 meters
- In scientific notation, this is approximately \( 8.85 \times 10^3 \) meters.
-
Mount Olympus:
- Height = 2,919.07 meters
- In scientific notation, this is approximately \( 2.92 \times 10^3 \) meters.
Next, we estimate the ratio:
\[ \text{Ratio} \approx \frac{8.85 \times 10^3}{2.92 \times 10^3} \]
We can simplify this by dividing the coefficients and cancelling out \(10^3\):
\[ \text{Ratio} \approx \frac{8.85}{2.92} \approx 3.03 \]
Thus, \( \text{Ratio} \approx 3.03 \).
The estimated ratio in scientific notation is approximately:
\[ 3.03 \times 10^0 \quad \text{(or simply, 3.03)} \]
Part B:
Now, we calculate the exact ratio using the actual heights:
\[ \text{Ratio} = \frac{\text{Height of Mount Everest}}{\text{Height of Mount Olympus}} = \frac{8849.87 \text{ m}}{2919.07 \text{ m}} \]
Using actual values, we perform the division:
\[ \text{Ratio} \approx \frac{8849.87}{2919.07} \approx 3.03 \]
To get the precise value, the calculation is:
\[ \text{Ratio} \approx 3.03 \]
Converting this into scientific notation:
\[ \text{Ratio} \approx 3.03 \times 10^0 \quad \text{(or simply, 3.03)} \]
Part C:
To conclude whether the estimate in Part A is reasonable, we compare both the estimated result and the calculated result.
In Part A, we estimated that Mount Everest is approximately \(3.03\) times greater than Mount Olympus. In Part B, we found the exact ratio to be roughly the same, \(3.03\). Since both the estimate (in Part A) and the calculation (in Part B) arrive at a similar value, we can conclude that the estimate in Part A is reasonable. This consistency between our rough estimate and the exact calculation demonstrates that our initial calculations were quite accurate.
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