7 5.8×10^7

Question

bacteria are measured to be in a dirt sample that weighs 11 gram. Use scientific notation to express the number of bacteria that would be in a sample weighing 18 grams.

Answer 1

To determine the number of bacteria in an 18-gram sample based on the information given for the 11-gram sample, we first need to calculate the bacteria density (bacteria per gram) in the 11-gram sample:

\[ \text{Bacteria density} = \frac{7.5 \times 10^7 \text{ bacteria}}{11 \text{ grams}} \approx 6.81818 \times 10^6 \text{ bacteria/gram} \]

Next, we can multiply this density by the weight of the new sample (18 grams) to find the total number of bacteria:

\[ \text{Total bacteria in 18 grams} = 6.81818 \times 10^6 \text{ bacteria/gram} \times 18 \text{ grams} = 1.227273 \times 10^8 \text{ bacteria} \]

To express this result in scientific notation, we round \(1.227273 \times 10^8\) to three significant figures:

\[ \approx 1.23 \times 10^8 \]

Thus, the number of bacteria in an 18-gram sample is approximately \(1.23 \times 10^8\).

Answer 2

The approximate average distances from the sun to Neptune and Venus are listed below:
Neptune:4.5×10^9
kilometers
Venus:1.08×10^8
kilometers
How many times farther from the sun is Neptune? Write your answer in standard notation, rounding to the nearest tenth

Answer 3

To determine how many times farther Neptune is from the Sun compared to Venus, we can divide the distance to Neptune by the distance to Venus.

Given:

• Distance to Neptune: \(4.5 \times 10^9\) kilometers
• Distance to Venus: \(1.08 \times 10^8\) kilometers

Now, let's perform the division:

\[ \text{Number of times farther} = \frac{4.5 \times 10^9 \text{ km}}{1.08 \times 10^8 \text{ km}} \]

We can simplify this by dividing the coefficients (4.5 and 1.08) and subtracting the exponents:

\[ = \frac{4.5}{1.08} \times 10^{9-8} \] \[ = \frac{4.5}{1.08} \times 10^{1} \] \[ = 4.16667 \times 10^{1} \]

Calculating the first part:

\[ \frac{4.5}{1.08} \approx 4.16667 \]

So,

\[ 4.16667 \times 10^{1} = 41.6667 \]

Now, rounding to the nearest tenth:

\[ 41.6667 \approx 41.7 \]

Thus, Neptune is approximately 41.7 times farther from the Sun than Venus.

Answer 4

A water park has pools, slides, and rides that, in total, make use of 4.6 x 10^5 gallons of water. They plan to add a ride that would make use of an additional 6.7 x 10^4 gallons of water. Use scientific notation to express the total gallons of water made use of in the park after the new ride is installed.

Answer 5

To find the total gallons of water used in the water park after adding a new ride, we need to add the initial amount of water (in scientific notation) to the additional amount of water (also in scientific notation).

Given:

• Initial water usage: \(4.6 \times 10^5\) gallons
• Additional water usage: \(6.7 \times 10^4\) gallons

Since these two quantities have different exponents (5 and 4), we can make the addition easier by expressing \(6.7 \times 10^4\) in terms of \(10^5\):

\[ 6.7 \times 10^4 = 0.67 \times 10^5 \]

Now we can add it to \(4.6 \times 10^5\):

\[ 4.6 \times 10^5 + 0.67 \times 10^5 = (4.6 + 0.67) \times 10^5 \]

Calculating the sum of the coefficients:

\[ 4.6 + 0.67 = 5.27 \]

So we now have:

\[ 5.27 \times 10^5 \]

This is already in scientific notation. Therefore, the total gallons of water used in the park after the new ride is installed is \(5.27 \times 10^5\) gallons.

Answer 6

The approximate areas of Colorado and Hawaii are listed below

Colorado: 2.7 x 10^5 square kilometers
Hawaii: 2.83 x 10^4 square kilometers

How much larger is Colorado? Express your answer using scientific notation.

Answer 7

To find out how much larger Colorado is compared to Hawaii, we'll subtract the area of Hawaii from the area of Colorado.

Given:

• Area of Colorado: \(2.7 \times 10^5\) square kilometers
• Area of Hawaii: \(2.83 \times 10^4\) square kilometers

Since the exponents are different (5 for Colorado and 4 for Hawaii), we will convert \(2.83 \times 10^4\) to the same exponent as Colorado's area:

\[ 2.83 \times 10^4 = 0.283 \times 10^5 \]

Now we can perform the subtraction:

\[ 2.7 \times 10^5 - 0.283 \times 10^5 = (2.7 - 0.283) \times 10^5 \]

Calculating the coefficient:

\[ 2.7 - 0.283 = 2.417 \]

Therefore, we have:

\[ 2.417 \times 10^5 \]

Now, we can express this in standard scientific notation. The result is already in the correct form.

Thus, Colorado is \( \mathbf{2.417 \times 10^5} \) square kilometers larger than Hawaii.

Answer 8

A water park has pools, slides, and rides that, in total, make use of 7.9 x 10^6 gallons of water. They plan to add a ride that would make use of an additional 6.7 x 10^4 gallons of water. Use scientific notation to express the total gallons of water made use of in the park after the new ride is installed.

Answer 9

To find the total gallons of water used in the water park after adding a new ride, we will add the current water usage to the additional water usage.

Given:

• Current water usage: \(7.9 \times 10^6\) gallons
• Additional water usage: \(6.7 \times 10^4\) gallons

Since these two numbers are expressed in scientific notation with different exponents (6 for the first and 4 for the second), we will convert \(6.7 \times 10^4\) to the same exponent as \(7.9 \times 10^6\) to make the addition easier.

We can convert \(6.7 \times 10^4\) to a number with \(10^6\) by expressing it as:

\[ 6.7 \times 10^4 = 0.067 \times 10^6 \]

Now we can add the two quantities:

\[ 7.9 \times 10^6 + 0.067 \times 10^6 = (7.9 + 0.067) \times 10^6 \]

Calculating the sum of the coefficients:

\[ 7.9 + 0.067 = 7.967 \]

So, we have:

\[ 7.967 \times 10^6 \]

This is already in scientific notation. Therefore, the total gallons of water used in the park after the new ride is installed is:

\[ \mathbf{7.967 \times 10^6} \text{ gallons} \]

Answer 10

A water park has pools, slides, and rides that, in total, make use of 7.5 x 10^6 gallons of water. They plan to add a ride that would make use of an additional 7.8 x 10^4 gallons of water. Use scientific notation to express the total gallons of water made use of in the park after the new ride is installed

Answer 11

To calculate the total gallons of water used in the water park after adding a new ride, we will add the current water usage to the additional water usage.

Given:

• Current water usage: \(7.5 \times 10^6\) gallons
• Additional water usage: \(7.8 \times 10^4\) gallons

Since the two quantities are expressed in scientific notation with different exponents (6 for the first and 4 for the second), we can convert \(7.8 \times 10^4\) to be expressed in terms of \(10^6\):

\[ 7.8 \times 10^4 = 0.078 \times 10^6 \]

Now we can add the two amounts:

\[ 7.5 \times 10^6 + 0.078 \times 10^6 = (7.5 + 0.078) \times 10^6 \]

Calculating the sum of the coefficients:

\[ 7.5 + 0.078 = 7.578 \]

So we have:

\[ 7.578 \times 10^6 \]

This is in proper scientific notation. Therefore, the total gallons of water made use of in the park after the new ride is installed is:

\[ \mathbf{7.578 \times 10^6} \text{ gallons} \]

Answer 12

A water park has pools, slides, and rides that, in total, make use of 4.1 x 10^7 gallons of water. They plan to add a ride that would make use of an additional 630,000 gallons of water. Use scientific notation to express the total gallons of water made use of in the park after the new ride is installed

Answer 13

To find the total gallons of water used in the water park after adding a new ride, we will add the current water usage to the additional water usage.

Given:

• Current water usage: \(4.1 \times 10^7\) gallons
• Additional water usage: 630,000 gallons

First, we need to express 630,000 in scientific notation:

\[ 630,000 = 6.3 \times 10^5 \text{ gallons} \]

Now we can add this to the current water usage. Since \(4.1 \times 10^7\) is in the order of \(10^7\), we can express \(6.3 \times 10^5\) in terms of \(10^7\):

\[ 6.3 \times 10^5 = 0.063 \times 10^7 \]

Now we can add the two amounts:

\[ 4.1 \times 10^7 + 0.063 \times 10^7 = (4.1 + 0.063) \times 10^7 \]

Calculating the sum of the coefficients:

\[ 4.1 + 0.063 = 4.163 \]

So, we have:

\[ 4.163 \times 10^7 \]

This is in proper scientific notation. Therefore, the total gallons of water made use of in the park after the new ride is installed is:

\[ \mathbf{4.163 \times 10^7} \text{ gallons} \]

Answer 14

The size of a cell is typically found by capturing an image under a microscope then using software to measure its diameter. Two cells are measured using this

Cell C: 7.76 x 10^-2 centimeters
Cell D: 8.4 x 10^-4 centimeters

How many times larger is the diameter of cell C than the diameter of cell D? Write your answer in standard notation, rounding to the nearest tenth

Answer 15

To find out how many times larger the diameter of Cell C is compared to Cell D, we will divide the diameter of Cell C by the diameter of Cell D.

Given:

• Diameter of Cell C: \(7.76 \times 10^{-2}\) centimeters
• Diameter of Cell D: \(8.4 \times 10^{-4}\) centimeters

Now we perform the division:

\[ \text{Number of times larger} = \frac{7.76 \times 10^{-2}}{8.4 \times 10^{-4}} \]

We can simplify this expression by dividing the coefficients and subtracting the exponents:

\[ = \frac{7.76}{8.4} \times 10^{-2 - (-4)} = \frac{7.76}{8.4} \times 10^{-2 + 4} = \frac{7.76}{8.4} \times 10^{2} \]

Calculating \(\frac{7.76}{8.4}\):

\[ \frac{7.76}{8.4} \approx 0.9226 \]

Then we multiply by \(10^2\):

\[ 0.9226 \times 10^2 = 92.26 \]

Finally, rounding this to the nearest tenth:

\[ 92.26 \approx 92.3 \]

Thus, the diameter of Cell C is approximately 92.3 times larger than the diameter of Cell D.

Answer 16

The size of a cell is typically found by capturing an image under a microscope then using software to measure its diameter. Two cells are measured using this

Cell C: 9.67 x 10^-3 centimeters
Cell D: 8.1 x 10^-4 centimeters

How many times larger is the diameter of cell C than the diameter of cell D? Write your answer in standard notation, rounding to the nearest tenth

Answer 17

To determine how many times larger the diameter of Cell C is compared to Cell D, we will divide the diameter of Cell C by the diameter of Cell D.

Given:

• Diameter of Cell C: \(9.67 \times 10^{-3}\) centimeters
• Diameter of Cell D: \(8.1 \times 10^{-4}\) centimeters

Now we perform the division:

\[ \text{Number of times larger} = \frac{9.67 \times 10^{-3}}{8.1 \times 10^{-4}} \]

We can simplify this by dividing the coefficients and subtracting the exponents:

\[ = \frac{9.67}{8.1} \times 10^{-3 - (-4)} = \frac{9.67}{8.1} \times 10^{-3 + 4} = \frac{9.67}{8.1} \times 10^{1} \]

Next, we calculate \(\frac{9.67}{8.1}\):

\[ \frac{9.67}{8.1} \approx 1.1938 \]

Now multiplying by \(10^1\):

\[ 1.1938 \times 10^{1} \approx 11.938 \]

Rounding to the nearest tenth:

\[ 11.938 \approx 11.9 \]

Thus, the diameter of Cell C is approximately 11.9 times larger than the diameter of Cell D.

Answer 18

A water park has pools, slides, and rides that, in total, make use of 8.1 x 10^7 gallons of water. They plan to add a ride that would make use of an additional 54,000 gallons of water. Use scientific notation to express the total gallons of water made use of in the park after the new ride is installed

Answer 19

To find the total gallons of water used in the water park after adding a new ride, we'll add the current water usage to the additional water usage.

Given:

• Current water usage: \(8.1 \times 10^7\) gallons
• Additional water usage: 54,000 gallons

First, we need to convert 54,000 into scientific notation:

\[ 54,000 = 5.4 \times 10^4 \text{ gallons} \]

Now we can perform the addition. Since \(8.1 \times 10^7\) is in the order of \(10^7\) and \(5.4 \times 10^4\) is in the order of \(10^4\), we will convert \(5.4 \times 10^4\) so that it has the same exponent as \(8.1 \times 10^7\):

\[ 5.4 \times 10^4 = 0.054 \times 10^7 \]

Next, we can add this to the current water usage:

\[ 8.1 \times 10^7 + 0.054 \times 10^7 = (8.1 + 0.054) \times 10^7 \]

Calculating the sum of the coefficients:

\[ 8.1 + 0.054 = 8.154 \]

Thus, we have:

\[ 8.154 \times 10^7 \]

This is now in proper scientific notation. Therefore, the total gallons of water made use of in the park after the new ride is installed is:

\[ \mathbf{8.154 \times 10^7} \text{ gallons} \]

Answer 20

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Answer 21

I apologize for any confusion. Let’s calculate the total gallons of water used in the water park again, step by step.

Given:

• Current water usage: \(8.1 \times 10^{7}\) gallons
• Additional water usage: 54,000 gallons

Step 1: Convert 54,000 to Scientific Notation

To write 54,000 in scientific notation: \[ 54,000 = 5.4 \times 10^{4} \]

Step 2: Align the Powers of 10 for Addition

Since \(8.1 \times 10^{7}\) is in \(10^7\), we can convert \(5.4 \times 10^{4}\) to have the same exponent:

\[ 5.4 \times 10^{4} = 0.054 \times 10^{7} \]

Step 3: Add the Two Values

Now we can add these two values:

\[ 8.1 \times 10^{7} + 0.054 \times 10^{7} = (8.1 + 0.054) \times 10^{7} \]

Calculating the sum of the coefficients:

\[ 8.1 + 0.054 = 8.154 \]

Final Result

Now, we combine the results:

\[ 8.154 \times 10^{7} \]

Thus, the total gallons of water made use of in the park after the new ride is installed is: \[ \mathbf{8.154 \times 10^{7}} \text{ gallons} \]

If this doesn't address your question, please let me know where the confusion lies so I can assist you more efficiently!

Answer 22

The distance from Earth to Mercury is 9.21 x 10^7 kilometers How long would it take a rocket, traveling at 5.72 x 10^4 kilometers per hour to travel from Earth to Mercury? Round your answer to the nearest whole number of hours

Answer 23

To determine how long it would take a rocket traveling at a speed of \(5.72 \times 10^4\) kilometers per hour to travel a distance of \(9.21 \times 10^7\) kilometers, we can use the formula:

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

Step 1: Calculate Time

We will first plug in the values for distance and speed:

\[ \text{Time} = \frac{9.21 \times 10^7 \text{ km}}{5.72 \times 10^4 \text{ km/h}} \]

Step 2: Divide the Coefficients

Divide the numerical coefficients:

\[ \frac{9.21}{5.72} \approx 1.607 \]

Step 3: Subtract the Exponents

Now we subtract the exponents:

\[ 10^{7 - 4} = 10^{3} \]

Step 4: Combine the Results

Putting it all together:

\[ \text{Time} \approx 1.607 \times 10^{3} \text{ hours} \]

Step 5: Convert to Standard Notation

Now convert \(1.607 \times 10^{3}\) to standard notation:

\[ 1.607 \times 10^{3} = 1607 \text{ hours} \]

Step 6: Round to the Nearest Whole Number

Rounding to the nearest whole number gives:

\[ \text{Time} \approx 1607 \text{ hours} \]

Thus, it would take approximately 1607 hours for the rocket to travel from Earth to Mercury.