Sure! Here are the solutions to the problems provided:
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For what value of a will 8 × 10^3 be 40 times larger than a × 10^2? \[ 8 \times 10^3 = 40 \times (a \times 10^2) \implies 8 \times 10^3 = 40a \times 10^2 \] Dividing both sides by \(10^2\): \[ 8 \times 10 = 40a \implies a = \frac{8 \times 10}{40} = \frac{80}{40} = 2 \] Answer: a = 2
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One second is approximately 0.0000115741 of a day. What is this number in scientific notation? \[ 0.0000115741 = 1.15741 \times 10^{-5} \] Answer: 1.15741 × 10^{-5}
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The temperature of the sun at its center is about 2.7 × 10^7 degrees Fahrenheit. What is this temperature written in decimal notation? \[ 2.7 \times 10^7 = 27,000,000 \] Answer: 27,000,000 degrees Fahrenheit
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Use scientific notation to determine which of these numbers has the least value: 98,789,000; 987,890,000; or 89,987,000.
- 98,789,000 = \(9.8789 \times 10^7\)
- 987,890,000 = \(9.8789 \times 10^8\)
- 89,987,000 = \(8.9987 \times 10^7\)
The least value is \(8.9987 \times 10^7\) (89,987,000). Answer: 8.9987 × 10^7
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Place the numbers in descending order.
- \(2.3 \times 10^{-8}\)
- \(3.2 \times 10^{-10}\)
- \(2.3 \times 10^{-10}\)
- \(2.3 \times 10^{-12}\)
Descending order: \(2.3 \times 10^{-8} ; 3.2 \times 10^{-10} ; 2.3 \times 10^{-10} ; 2.3 \times 10^{-12}\)
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Which distance uses the most appropriate units?
- \(3 \times 10^2\) kilometers
- \(3 \times 10^5\) meters
- \(3 \times 10^7\) centimeters
- \(3 \times 10^8\) millimeters
The most appropriate would likely be kilometers, since it represents larger distances in normal practical usage. Answer: 3 × 10^2 kilometers
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A national park covers an area of 1.42×10^6 acres. A nearby national forest covers an area of 1.2×10^6 acres. What is the total number of acres in the park and the forest combined? \[ 1.42 \times 10^6 + 1.2 \times 10^6 = (1.42 + 1.2) \times 10^6 = 2.62 \times 10^6 \] Answer: 2.62 × 10^6 acres
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The distance from Venus to the sun is about 67,000,000 miles. The distance from Earth to the sun is about 2.6×10^7 farther. What is the distance from Earth to the sun? \[ 67,000,000 + 2.6 \times 10^7 = 6.7 \times 10^7 + 2.6 \times 10^7 = 9.3 \times 10^7 \] Answer: 9.3 × 10^7 miles
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Calculate 4.8×10−3 − 9.4×10−6. First, express \(4.8 \times 10^{-3}\) in terms of \(10^{-6}\): \[ 4.8 \times 10^{-3} = 4800 \times 10^{-6} \] Now, perform the subtraction: \[ 4800 \times 10^{-6} - 9.4 \times 10^{-6} = (4800 - 9.4) \times 10^{-6} \approx 4790.6 \times 10^{-6} = 4.7906 \times 10^{-3} \] Answer: 4.79 × 10^{-3}
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A strand of hair has an average mass of 2.17×10−5 ounces. A grain of rice has an average mass of 0.002 ounces. How much heavier is a grain of rice than a strand of hair? \[ 0.002 = 2.0 \times 10^{-3} \text{ ounces} \] Now subtract: \[ 2.0 \times 10^{-3} - 2.17 \times 10^{-5} \approx 2.0 \times 10^{-3} = 2.0 \times 10^{-3} \text{ ounces} \] (since \(2.17 \times 10^{-5}\) is negligible at this scale). Answer: 2.00 × 10^{-3} ounces
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Light travels at an approximate rate of 1.9×10^5 mi./sec. There are about 3.2×10^7 seconds in a year. About how many miles does light travel in a year? \[ 1.9 \times 10^5 \times 3.2 \times 10^7 = (1.9 \times 3.2) \times (10^5 \times 10^7) = 6.08 \times 10^{12} \] Answer: 6.08 × 10^{12} miles
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The average tectonic plate moves at the rate of 0.006 m per year. How many meters would it move in 2×10^6 years? \[ 0.006 \times 2 \times 10^6 = 1.2 \times 10^4 \] Answer: 1.2 × 10^4 m
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An asteroid is heading toward Mars at the rate of 7.7×10^4 mi./hr. If the asteroid is 3.311×10^8 miles away from Mars, how many hours will it be before it hits Mars? \[ \frac{3.311 \times 10^8}{7.7 \times 10^4} = \frac{3.311}{7.7} \times 10^{8-4} \approx 4.2961 \times 10^4 \] Answer: 4.30 × 10^4 hours
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A field of 3.9 mi^2 is being subdivided into 1.3×10^3 family gardens. How many square miles will there be in each garden? \[ \frac{3.9 , \text{mi}^2}{1.3 \times 10^3} \approx 0.003 ,\text{mi}^2 = 3.0 \times 10^{-3} ,\text{mi}^2 \] Answer: 3.0 × 10^{-3} mi^2
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