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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? Write the answer in scientific notation. Express the answer to the exact decimal place

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To find how many miles light travels in one year, we can multiply the speed of light by the number of seconds in a year.

The speed of light is approximately \( 1.9 \times 10^5 \) miles per second and there are about \( 3.2 \times 10^7 \) seconds in a year.

We can set up the calculation as follows:

\[ \text{Distance} = (\text{Speed}) \times (\text{Time}) = (1.9 \times 10^5 , \text{mi/sec}) \times (3.2 \times 10^7 , \text{sec}) \]

Now, we can multiply the coefficients and the powers of ten separately:

  1. Multiply the coefficients: \[ 1.9 \times 3.2 = 6.08 \]

  2. Add the exponents (since we are multiplying powers of ten): \[ 10^5 \times 10^7 = 10^{(5+7)} = 10^{12} \]

Now, we combine these results:

\[ \text{Distance} = 6.08 \times 10^{12} , \text{miles} \]

Thus, the distance that light travels in a year, expressed in scientific notation, is \( \boxed{6.08 \times 10^{12}} \) miles.

The average tectonic plate moves at hte 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

To find how many meters a tectonic plate would move in \( 2 \times 10^6 \) years, we can multiply the average movement per year by the number of years.

The average movement rate of a tectonic plate is \( 0.006 , \text{m/year} \), and we need to calculate the distance it would move over \( 2 \times 10^6 \) years.

We can set up the calculation as follows:

\[ \text{Distance} = (\text{Rate}) \times (\text{Time}) = (0.006 , \text{m/year}) \times (2 \times 10^6 , \text{years}) \]

Now, multiply the numbers:

  1. Calculate the coefficient: \[ 0.006 \times 2 = 0.012 \]

  2. Add the exponent from the scientific notation: \[ 10^0 \times 10^{6} = 10^{(0 + 6)} = 10^{6} \]

Combining those results, we have: \[ \text{Distance} = 0.012 \times 10^{6} , \text{m} \]

To express \( 0.012 \) in proper scientific notation, we can write it as: \[ 0.012 = 1.2 \times 10^{-2} \]

Now substituting this back into our distance calculation: \[ \text{Distance} = (1.2 \times 10^{-2}) \times 10^{6} , \text{m} = 1.2 \times 10^{4} , \text{m} \]

Thus, the distance that tectonic plates would move in \( 2 \times 10^6 \) years, expressed in scientific notation, is \( \boxed{1.2 \times 10^4} \) meters.

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? Write the answer in scientific notation, expressed to the exact decimal place.

To find out how many hours it will take for the asteroid to hit Mars, we can use the formula:

\[ \text{Time} = \frac{\text{Distance}}{\text{Rate}} \]

Given:

  • Distance to Mars: \( 3.311 \times 10^8 \) miles
  • Rate of the asteroid: \( 7.7 \times 10^4 \) miles per hour

Plugging in the values:

\[ \text{Time} = \frac{3.311 \times 10^8 , \text{mi}}{7.7 \times 10^4 , \text{mi/hr}} \]

To perform this division, we can divide the coefficients and then subtract the exponents of the powers of ten:

  1. Divide the coefficients: \[ \frac{3.311}{7.7} \approx 0.430 \]

  2. Subtract the exponents: \[ 10^{8} \div 10^{4} = 10^{(8-4)} = 10^{4} \]

Now combine these results:

\[ \text{Time} \approx 0.430 \times 10^{4} , \text{hours} \]

To express \( 0.430 \) in proper scientific notation, we can write it as:

\[ 0.430 = 4.30 \times 10^{-1} \]

Now substituting this back into our time calculation: \[ \text{Time} = (4.30 \times 10^{-1}) \times 10^{4} = 4.30 \times 10^{3} , \text{hours} \]

Thus, the time it will take for the asteroid to hit Mars, expressed in scientific notation, is \( \boxed{4.30 \times 10^3} \) hours.

A field of 3.9 mi2 is being divided into 1.3 × 103 family gardens. How many square miles will there be in each garden? Write the answer in scientific notation

To find out how many square miles will be in each family garden, we need to divide the total area of the field by the number of family gardens.

Given:

  • Total area of the field: \( 3.9 , \text{mi}^2 \)
  • Number of family gardens: \( 1.3 \times 10^3 \)

The calculation can be set up as follows:

\[ \text{Area per garden} = \frac{\text{Total area}}{\text{Number of gardens}} = \frac{3.9 , \text{mi}^2}{1.3 \times 10^3} \]

Now we can perform the division:

  1. Divide the coefficients: \[ \frac{3.9}{1.3} \approx 3.0 \]

  2. Subtract the exponents: \[ 10^{0} \div 10^{3} = 10^{(0-3)} = 10^{-3} \]

Combining these results gives us:

\[ \text{Area per garden} \approx 3.0 \times 10^{-3} , \text{mi}^2 \]

To express \( 3.0 \) in proper scientific notation, we simply keep it as \( 3.0 \).

Thus, the area in each family garden, expressed in scientific notation, is \( \boxed{3.0 \times 10^{-3}} , \text{mi}^2 \).

The top speed of a Rocket Sled is 10,326 kilometers per hours. Is this faster or slower than the Autonomous Drone?

For full credit:

1) In full sentences, describe the steps you took to convert the Rocket Sled's speed into scientific notation.

2) You must include the correct Scientific Notation for the Rocket Sled to earn full points.)

3) Compare the speed to the Autonomous Drone. Explain how you know it is faster or slower.

sports car (4.15) • 10^2
Apollo command and service module (3.99) • 10^4
Jet boat (5.1) • 10^2
Autonomous drone (2.1) • 10^4

To compare the speed of the Rocket Sled to the speed of the Autonomous Drone, I first converted the Rocket Sled's speed from standard form into scientific notation.

  1. Steps for Conversion: a. The speed of the Rocket Sled is given as \(10,326\) kilometers per hour. To convert this number into scientific notation, I first identified the significant digits, which are \(1.0326\). b. Next, I noted that \(10,326\) can be expressed as \(1.0326 \times 10^4\). This is because moving the decimal point four places to the left gives us \(1.0326\), thus we multiply by \(10^4\) to represent the original number. c. Therefore, the speed of the Rocket Sled in scientific notation is \(1.0326 \times 10^4\) kilometers per hour.

  2. Scientific Notation of the Rocket Sled: The speed of the Rocket Sled in scientific notation is \( \boxed{1.0326 \times 10^4} \) kilometers per hour.

  3. Comparison to Autonomous Drone: The speed of the Autonomous Drone is given as \(2.1 \times 10^4\) kilometers per hour. To compare the two speeds, I looked at the coefficients in their scientific notation forms. The coefficient of the Rocket Sled is \(1.0326\), while the coefficient of the Autonomous Drone is \(2.1\). Since \(1.0326\) is less than \(2.1\), it is evident that \(1.0326 \times 10^4\) is slower than \(2.1 \times 10^4\). Thus, I conclude that the Rocket Sled is slower than the Autonomous Drone.