How many times larger is 4 × 10^12 than 2 × 10^9?

Question

A.
200
B.
2,000
C.
20,000
D.
20

Answer 1

To determine how many times larger \( 4 \times 10^{12} \) is than \( 2 \times 10^{9} \), we divide the two quantities.

\[ \frac{4 \times 10^{12}}{2 \times 10^{9}} = \frac{4}{2} \times \frac{10^{12}}{10^{9}} \]

Calculating the first part:

\[ \frac{4}{2} = 2 \]

For the second part, we use the rule of exponents \( \frac{10^{a}}{10^{b}} = 10^{a-b} \):

\[ \frac{10^{12}}{10^{9}} = 10^{12-9} = 10^{3} \]

Now, combine both parts:

\[ 2 \times 10^{3} = 2 \times 1000 = 2000 \]

Thus, \( 4 \times 10^{12} \) is 2000 times larger than \( 2 \times 10^{9} \).

The answer is \( \text{B. } 2,000 \).

Answer 2

Which of the following is a reasonable estimate for the weight of a U.S. penny?
A.
5 × 10^1 lb
B.
5 × 10^-1 lb
C.
5 × 10^-3 lb
D.
5 × 10^3 lb

Answer 3

To estimate the weight of a U.S. penny, we can consider its actual weight. A U.S. penny weighs approximately 2.5 grams.

To convert grams to pounds, we use the conversion factor where 1 pound is approximately 454 grams.

So,

\[ \text{Weight in pounds} \approx \frac{2.5 \text{ grams}}{454 \text{ grams/pound}} \approx 0.00551 \text{ pounds} \]

Now, we can express this weight in scientific notation:

\[ 0.00551 \text{ pounds} \approx 5.51 \times 10^{-3} \text{ pounds} \]

Therefore, a reasonable estimate for the weight of a U.S. penny in scientific notation is roughly \( 5 \times 10^{-3} \) pounds.

Thus, the correct answer is:

C. \( 5 \times 10^{-3} \) lb.

Answer 4

A scientist measures the diameter of a bacteria cell and yeast cell. She finds the diameter of a bacteria cell to be 0.0000021 meters and the diameter of a yeast cell to be 0.00000398 meters.

The approximate diameter of the bacteria cell is represented by point___ on the number line.
The approximate diameter of the yeast cell is represented by point____ on the number line.
The diameter of the bacteria cell is approximately___ times smaller than the diameter of the yeast cell.

Answer 5

Let's first convert the diameters from meters to scientific notation for easier comparison:

1. Diameter of the bacteria cell: \[ 0.0000021 \text{ meters} = 2.1 \times 10^{-6} \text{ meters} \]

2. Diameter of the yeast cell: \[ 0.00000398 \text{ meters} = 3.98 \times 10^{-6} \text{ meters} \]

Number Line Representation:

• The approximate diameter of the bacteria cell \( 2.1 \times 10^{-6} \) meters would be represented at point approximately 2.1 on the number line of \( 10^{-6} \) meters.
• The approximate diameter of the yeast cell \( 3.98 \times 10^{-6} \) meters would be represented at point approximately 3.98 on the number line of \( 10^{-6} \) meters.

Calculating How Many Times Smaller: To find out how many times smaller the diameter of the bacteria cell is compared to the yeast cell, we divide the diameter of the yeast cell by the diameter of the bacteria cell:

\[ \text{Ratio} = \frac{\text{Diameter of yeast cell}}{\text{Diameter of bacteria cell}} = \frac{3.98 \times 10^{-6}}{2.1 \times 10^{-6}} \]

Cancelling out \( 10^{-6} \):

\[ \text{Ratio} = \frac{3.98}{2.1} \]

Calculating the ratio:

\[ \text{Ratio} \approx 1.8952 \]

So, the diameter of the bacteria cell is approximately 1.9 times smaller than the diameter of the yeast cell.

Final Answers:

• The approximate diameter of the bacteria cell is represented by point 2.1 on the number line.
• The approximate diameter of the yeast cell is represented by point 3.98 on the number line.
• The diameter of the bacteria cell is approximately 1.9 times smaller than the diameter of the yeast cell.