Question
Scientific notation unit test.
10. 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? Express your answer in scientific notation.
11. Light travels at an approximate rate of 1.9×105 mi./sec. There are about 3.2×107 seconds in a year. About how many miles does light travel in a year? Write the answer in scientific notation. Express the answer to the exact decimal place.
12.The average tectonic plate moves at the rate of 0.006 m per year. How many meters would it move in 2×106 years? Write the answer in scientific notation, expressed to the exact decimal place.
13. An asteroid is heading toward Mars at the rate of 7.7×104 mi./hr. If the asteroid is 3.311×108 miles away from Mars, how many hours will it be before it hits Mars? Write the answer in scientific notation, expressed to the exact decimal place.
14. A field of 3.9 mi2 is being subdivided into 1.3×103 family gardens. How many square miles will there be in each garden? Write the answer in scientific notation.
15. Charlee is using her calculator to find the product of two large numbers. Her calculator shows 1.78E12 on the display. Interpret what this means.
10. 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? Express your answer in scientific notation.
11. Light travels at an approximate rate of 1.9×105 mi./sec. There are about 3.2×107 seconds in a year. About how many miles does light travel in a year? Write the answer in scientific notation. Express the answer to the exact decimal place.
12.The average tectonic plate moves at the rate of 0.006 m per year. How many meters would it move in 2×106 years? Write the answer in scientific notation, expressed to the exact decimal place.
13. An asteroid is heading toward Mars at the rate of 7.7×104 mi./hr. If the asteroid is 3.311×108 miles away from Mars, how many hours will it be before it hits Mars? Write the answer in scientific notation, expressed to the exact decimal place.
14. A field of 3.9 mi2 is being subdivided into 1.3×103 family gardens. How many square miles will there be in each garden? Write the answer in scientific notation.
15. Charlee is using her calculator to find the product of two large numbers. Her calculator shows 1.78E12 on the display. Interpret what this means.
Answers
GPT-4o
Sure, let's go through each question one by one and solve them.
### Question 10
A strand of hair has an average mass of \(2.17 \times 10^{-5}\) ounces and 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? Express your answer in scientific notation.
**Solution:**
1. Convert 0.002 ounces to scientific notation:
\[
0.002 = 2 \times 10^{-3}
\]
2. Find the difference between the masses:
\[
(2 \times 10^{-3}) - (2.17 \times 10^{-5})
\]
To subtract, convert \(2 \times 10^{-3}\) to the same exponent as \(2.17 \times 10^{-5}\):
\[
2 \times 10^{-3} = 200 \times 10^{-5}
\]
Now perform the subtraction:
\[
(200 \times 10^{-5}) - (2.17 \times 10^{-5}) = (200 - 2.17) \times 10^{-5} = 197.83 \times 10^{-5}
\]
Convert \(197.83 \times 10^{-5}\) to scientific notation:
\[
1.9783 \times 10^{-3}
\]
So, a grain of rice is \(\boxed{1.9783 \times 10^{-3}}\) ounces heavier than a strand of hair.
### Question 11
Light travels at an approximate rate of \(1.9 \times 10^5\) mi./sec. There are about \(3.2 \times 10^7\) seconds in a year. About how many miles does light travel in a year? Write the answer in scientific notation.
**Solution:**
Multiply the speed of light by the number of seconds in a year:
\[
(1.9 \times 10^5 \text{ mi/sec}) \times (3.2 \times 10^7 \text{ sec}) = 1.9 \times 3.2 \times 10^{5+7} = 6.08 \times 10^{12} \text{ mi}
\]
So, light travels \(\boxed{6.08 \times 10^{12}}\) miles in a year.
### Question 12
The average tectonic plate moves at the rate of 0.006 m per year. How many meters would it move in \(2 \times 10^6\) years? Write the answer in scientific notation, expressed to the exact decimal place.
**Solution:**
Multiply the rate of movement by the number of years:
\[
0.006 \text{ m/year} \times 2 \times 10^6 \text{ years} = 6 \times 10^{-3} \times 2 \times 10^6 = 12 \times 10^{3} = 1.2 \times 10^{4} \text{ m}
\]
So, the tectonic plate would move \(\boxed{1.2 \times 10^4}\) meters in \(2 \times 10^6\) years.
### Question 13
An asteroid is heading toward Mars at the rate of \(7.7 \times 10^4\) mi./hr. If the asteroid is \(3.311 \times 10^8\) miles away from Mars, how many hours will it be before it hits Mars? Write the answer in scientific notation, expressed to the exact decimal place.
**Solution:**
Divide the distance by the speed to find the time:
\[
\frac{3.311 \times 10^8 \text{ mi}}{7.7 \times 10^4 \text{ mi/hr}} = \frac{3.311}{7.7} \times 10^{8-4} = 0.4298701 \times 10^4 = 4.298701 \times 10^3 \text{ hr}
\]
So, it will take \(\boxed{4.298701 \times 10^3}\) hours for the asteroid to hit Mars.
### Question 14
A field of \(3.9\) mi² is being subdivided into \(1.3 \times 10^3\) family gardens. How many square miles will there be in each garden? Write the answer in scientific notation.
**Solution:**
Divide the total area by the number of gardens:
\[
\frac{3.9 \text{ mi}^2}{1.3 \times 10^3} = \frac{3.9}{1.3} \times 10^{-3} = 3 \times 10^{-3} \text{ mi}^2
\]
So, each garden will be \(\boxed{3 \times 10^{-3}}\) square miles.
### Question 15
Charlee is using her calculator to find the product of two large numbers. Her calculator shows 1.78E12 on the display. Interpret what this means.
**Solution:**
The display \(1.78E12\) means \(1.78 \times 10^{12}\).
So, Charlee’s calculator displays \(\boxed{1.78 \times 10^{12}}\).
### Question 10
A strand of hair has an average mass of \(2.17 \times 10^{-5}\) ounces and 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? Express your answer in scientific notation.
**Solution:**
1. Convert 0.002 ounces to scientific notation:
\[
0.002 = 2 \times 10^{-3}
\]
2. Find the difference between the masses:
\[
(2 \times 10^{-3}) - (2.17 \times 10^{-5})
\]
To subtract, convert \(2 \times 10^{-3}\) to the same exponent as \(2.17 \times 10^{-5}\):
\[
2 \times 10^{-3} = 200 \times 10^{-5}
\]
Now perform the subtraction:
\[
(200 \times 10^{-5}) - (2.17 \times 10^{-5}) = (200 - 2.17) \times 10^{-5} = 197.83 \times 10^{-5}
\]
Convert \(197.83 \times 10^{-5}\) to scientific notation:
\[
1.9783 \times 10^{-3}
\]
So, a grain of rice is \(\boxed{1.9783 \times 10^{-3}}\) ounces heavier than a strand of hair.
### Question 11
Light travels at an approximate rate of \(1.9 \times 10^5\) mi./sec. There are about \(3.2 \times 10^7\) seconds in a year. About how many miles does light travel in a year? Write the answer in scientific notation.
**Solution:**
Multiply the speed of light by the number of seconds in a year:
\[
(1.9 \times 10^5 \text{ mi/sec}) \times (3.2 \times 10^7 \text{ sec}) = 1.9 \times 3.2 \times 10^{5+7} = 6.08 \times 10^{12} \text{ mi}
\]
So, light travels \(\boxed{6.08 \times 10^{12}}\) miles in a year.
### Question 12
The average tectonic plate moves at the rate of 0.006 m per year. How many meters would it move in \(2 \times 10^6\) years? Write the answer in scientific notation, expressed to the exact decimal place.
**Solution:**
Multiply the rate of movement by the number of years:
\[
0.006 \text{ m/year} \times 2 \times 10^6 \text{ years} = 6 \times 10^{-3} \times 2 \times 10^6 = 12 \times 10^{3} = 1.2 \times 10^{4} \text{ m}
\]
So, the tectonic plate would move \(\boxed{1.2 \times 10^4}\) meters in \(2 \times 10^6\) years.
### Question 13
An asteroid is heading toward Mars at the rate of \(7.7 \times 10^4\) mi./hr. If the asteroid is \(3.311 \times 10^8\) miles away from Mars, how many hours will it be before it hits Mars? Write the answer in scientific notation, expressed to the exact decimal place.
**Solution:**
Divide the distance by the speed to find the time:
\[
\frac{3.311 \times 10^8 \text{ mi}}{7.7 \times 10^4 \text{ mi/hr}} = \frac{3.311}{7.7} \times 10^{8-4} = 0.4298701 \times 10^4 = 4.298701 \times 10^3 \text{ hr}
\]
So, it will take \(\boxed{4.298701 \times 10^3}\) hours for the asteroid to hit Mars.
### Question 14
A field of \(3.9\) mi² is being subdivided into \(1.3 \times 10^3\) family gardens. How many square miles will there be in each garden? Write the answer in scientific notation.
**Solution:**
Divide the total area by the number of gardens:
\[
\frac{3.9 \text{ mi}^2}{1.3 \times 10^3} = \frac{3.9}{1.3} \times 10^{-3} = 3 \times 10^{-3} \text{ mi}^2
\]
So, each garden will be \(\boxed{3 \times 10^{-3}}\) square miles.
### Question 15
Charlee is using her calculator to find the product of two large numbers. Her calculator shows 1.78E12 on the display. Interpret what this means.
**Solution:**
The display \(1.78E12\) means \(1.78 \times 10^{12}\).
So, Charlee’s calculator displays \(\boxed{1.78 \times 10^{12}}\).
u sure bout all that?
GPT-4o
Let's go through each problem again carefully to ensure correctness.
### Question 10
A strand of hair has an average mass of \(2.17 \times 10^{-5}\) 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? Express your answer in scientific notation.
**Solution:**
1. Convert 0.002 ounces to scientific notation:
\[
0.002 = 2 \times 10^{-3}
\]
2. Find the difference between the masses:
\[
(2 \times 10^{-3}) - (2.17 \times 10^{-5})
\]
To subtract, convert \(2 \times 10^{-3}\) to the same exponent as \(2.17 \times 10^{-5}\):
\[
2 \times 10^{-3} = 200 \times 10^{-5}
\]
Now perform the subtraction:
\[
(200 \times 10^{-5}) - (2.17 \times 10^{-5}) = (200 - 2.17) \times 10^{-5} = 197.83 \times 10^{-5}
\]
Convert \(197.83 \times 10^{-5}\) to scientific notation:
\[
1.9783 \times 10^{-3}
\]
So, a grain of rice is \(\boxed{1.9783 \times 10^{-3}}\) ounces heavier than a strand of hair.
### Question 11
Light travels at an approximate rate of \(1.9 \times 10^5\) mi./sec. There are about \(3.2 \times 10^7\) seconds in a year. About how many miles does light travel in a year? Write the answer in scientific notation.
**Solution:**
Multiply the speed of light by the number of seconds in a year:
\[
(1.9 \times 10^5 \text{ mi/sec}) \times (3.2 \times 10^7 \text{ sec}) = 1.9 \times 3.2 \times 10^{5+7} = 6.08 \times 10^{12} \text{ mi}
\]
So, light travels \(\boxed{6.08 \times 10^{12}}\) miles in a year.
### Question 12
The average tectonic plate moves at the rate of 0.006 m per year. How many meters would it move in \(2 \times 10^6\) years? Write the answer in scientific notation, expressed to the exact decimal place.
**Solution:**
Multiply the rate of movement by the number of years:
\[
0.006 \text{ m/year} \times 2 \times 10^6 \text{ years} = 6 \times 10^{-3} \times 2 \times 10^6 = 12 \times 10^{3} = 1.2 \times 10^{4} \text{ m}
\]
So, the tectonic plate would move \(\boxed{1.2 \times 10^4}\) meters in \(2 \times 10^6\) years.
### Question 13
An asteroid is heading toward Mars at the rate of \(7.7 \times 10^4\) mi./hr. If the asteroid is \(3.311 \times 10^8\) miles away from Mars, how many hours will it be before it hits Mars? Write the answer in scientific notation, expressed to the exact decimal place.
**Solution:**
Divide the distance by the speed to find the time:
\[
\frac{3.311 \times 10^8 \text{ mi}}{7.7 \times 10^4 \text{ mi/hr}} = \frac{3.311}{7.7} \times 10^{8-4} = 0.4298701 \times 10^4 = 4.298701 \times 10^3 \text{ hr}
\]
So, it will take \(\boxed{4.298701 \times 10^3}\) hours for the asteroid to hit Mars.
### Question 14
A field of \(3.9\) mi² is being subdivided into \(1.3 \times 10^3\) family gardens. How many square miles will there be in each garden? Write the answer in scientific notation.
**Solution:**
Divide the total area by the number of gardens:
\[
\frac{3.9 \text{ mi}^2}{1.3 \times 10^3} = \frac{3.9}{1.3} \times 10^{-3} = 3 \times 10^{-3} \text{ mi}^2
\]
So, each garden will be \(\boxed{3 \times 10^{-3}}\) square miles.
### Question 15
Charlee is using her calculator to find the product of two large numbers. Her calculator shows 1.78E12 on the display. Interpret what this means.
**Solution:**
The display \(1.78E12\) means \(1.78 \times 10^{12}\).
So, Charlee’s calculator displays \(\boxed{1.78 \times 10^{12}}\).
After revisiting the solutions, the calculations appear to be correct. Please let me know if you have any further questions or doubts!
### Question 10
A strand of hair has an average mass of \(2.17 \times 10^{-5}\) 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? Express your answer in scientific notation.
**Solution:**
1. Convert 0.002 ounces to scientific notation:
\[
0.002 = 2 \times 10^{-3}
\]
2. Find the difference between the masses:
\[
(2 \times 10^{-3}) - (2.17 \times 10^{-5})
\]
To subtract, convert \(2 \times 10^{-3}\) to the same exponent as \(2.17 \times 10^{-5}\):
\[
2 \times 10^{-3} = 200 \times 10^{-5}
\]
Now perform the subtraction:
\[
(200 \times 10^{-5}) - (2.17 \times 10^{-5}) = (200 - 2.17) \times 10^{-5} = 197.83 \times 10^{-5}
\]
Convert \(197.83 \times 10^{-5}\) to scientific notation:
\[
1.9783 \times 10^{-3}
\]
So, a grain of rice is \(\boxed{1.9783 \times 10^{-3}}\) ounces heavier than a strand of hair.
### Question 11
Light travels at an approximate rate of \(1.9 \times 10^5\) mi./sec. There are about \(3.2 \times 10^7\) seconds in a year. About how many miles does light travel in a year? Write the answer in scientific notation.
**Solution:**
Multiply the speed of light by the number of seconds in a year:
\[
(1.9 \times 10^5 \text{ mi/sec}) \times (3.2 \times 10^7 \text{ sec}) = 1.9 \times 3.2 \times 10^{5+7} = 6.08 \times 10^{12} \text{ mi}
\]
So, light travels \(\boxed{6.08 \times 10^{12}}\) miles in a year.
### Question 12
The average tectonic plate moves at the rate of 0.006 m per year. How many meters would it move in \(2 \times 10^6\) years? Write the answer in scientific notation, expressed to the exact decimal place.
**Solution:**
Multiply the rate of movement by the number of years:
\[
0.006 \text{ m/year} \times 2 \times 10^6 \text{ years} = 6 \times 10^{-3} \times 2 \times 10^6 = 12 \times 10^{3} = 1.2 \times 10^{4} \text{ m}
\]
So, the tectonic plate would move \(\boxed{1.2 \times 10^4}\) meters in \(2 \times 10^6\) years.
### Question 13
An asteroid is heading toward Mars at the rate of \(7.7 \times 10^4\) mi./hr. If the asteroid is \(3.311 \times 10^8\) miles away from Mars, how many hours will it be before it hits Mars? Write the answer in scientific notation, expressed to the exact decimal place.
**Solution:**
Divide the distance by the speed to find the time:
\[
\frac{3.311 \times 10^8 \text{ mi}}{7.7 \times 10^4 \text{ mi/hr}} = \frac{3.311}{7.7} \times 10^{8-4} = 0.4298701 \times 10^4 = 4.298701 \times 10^3 \text{ hr}
\]
So, it will take \(\boxed{4.298701 \times 10^3}\) hours for the asteroid to hit Mars.
### Question 14
A field of \(3.9\) mi² is being subdivided into \(1.3 \times 10^3\) family gardens. How many square miles will there be in each garden? Write the answer in scientific notation.
**Solution:**
Divide the total area by the number of gardens:
\[
\frac{3.9 \text{ mi}^2}{1.3 \times 10^3} = \frac{3.9}{1.3} \times 10^{-3} = 3 \times 10^{-3} \text{ mi}^2
\]
So, each garden will be \(\boxed{3 \times 10^{-3}}\) square miles.
### Question 15
Charlee is using her calculator to find the product of two large numbers. Her calculator shows 1.78E12 on the display. Interpret what this means.
**Solution:**
The display \(1.78E12\) means \(1.78 \times 10^{12}\).
So, Charlee’s calculator displays \(\boxed{1.78 \times 10^{12}}\).
After revisiting the solutions, the calculations appear to be correct. Please let me know if you have any further questions or doubts!