Question
There are about 6×10^19 atoms in a grain of sand. A cubic foot of sand can hold about 4×10^9 grains of sand of average size. Using those figures, how many atoms would you expect to find in a cubic foot of sand? Write the answer in scientific notation.
Answers
GPT-4o mini
To find the number of atoms in a cubic foot of sand, we can multiply the number of atoms in a single grain of sand by the number of grains of sand in a cubic foot.
Given:
- Number of atoms in a grain of sand = \( 6 \times 10^{19} \)
- Number of grains in a cubic foot of sand = \( 4 \times 10^{9} \)
Now, we can perform the multiplication:
\[
\text{Total atoms} = \text{Number of atoms in a grain} \times \text{Number of grains}
\]
\[
\text{Total atoms} = (6 \times 10^{19}) \times (4 \times 10^{9})
\]
Using the properties of multiplication in scientific notation, we can separate the coefficients and the powers of 10:
\[
\text{Total atoms} = (6 \times 4) \times (10^{19} \times 10^{9})
\]
Calculating the coefficients:
\[
6 \times 4 = 24
\]
For the powers of ten, we add the exponents:
\[
10^{19} \times 10^{9} = 10^{19+9} = 10^{28}
\]
Now combining these results gives:
\[
\text{Total atoms} = 24 \times 10^{28}
\]
To express this in scientific notation, we convert \( 24 \) into \( 2.4 \times 10^{1} \):
\[
24 \times 10^{28} = (2.4 \times 10^{1}) \times 10^{28} = 2.4 \times 10^{29}
\]
Thus, the number of atoms expected in a cubic foot of sand is:
\[
\boxed{2.4 \times 10^{29}}
\]
Given:
- Number of atoms in a grain of sand = \( 6 \times 10^{19} \)
- Number of grains in a cubic foot of sand = \( 4 \times 10^{9} \)
Now, we can perform the multiplication:
\[
\text{Total atoms} = \text{Number of atoms in a grain} \times \text{Number of grains}
\]
\[
\text{Total atoms} = (6 \times 10^{19}) \times (4 \times 10^{9})
\]
Using the properties of multiplication in scientific notation, we can separate the coefficients and the powers of 10:
\[
\text{Total atoms} = (6 \times 4) \times (10^{19} \times 10^{9})
\]
Calculating the coefficients:
\[
6 \times 4 = 24
\]
For the powers of ten, we add the exponents:
\[
10^{19} \times 10^{9} = 10^{19+9} = 10^{28}
\]
Now combining these results gives:
\[
\text{Total atoms} = 24 \times 10^{28}
\]
To express this in scientific notation, we convert \( 24 \) into \( 2.4 \times 10^{1} \):
\[
24 \times 10^{28} = (2.4 \times 10^{1}) \times 10^{28} = 2.4 \times 10^{29}
\]
Thus, the number of atoms expected in a cubic foot of sand is:
\[
\boxed{2.4 \times 10^{29}}
\]