leah is studying a bacteria sample that increases 4 times in mass every day. at the end of day 1 of her…
leah is studying a bacteria sample that increases 4 times in mass every day. at the end of day 1 of her study, the sample has a mass of 4 milligrams. what is the mass of the bacteria sample after 6 days of study? write each expression from the box in the correct column of the table. \\(4^6\\) \\(6^4\\) \\(4\\times6\\) \\(4\\times4\\times4\\times4\\times4\\times4\\) \\(6\\times6\\times6\\times6\\) \\(16\\times256\\) \\(\\begin{array}{|c|c|}\\hline\\text{represents the mass, in milligrams, of the bacteria sample after 6 days}&\\\\\\hline\\text{does not represent the mass, in milligrams, of the bacteria sample after 6 days}&\\\\\\hline\\end{array}\\)
Answer
Explanation:
The mass of the bacteria sample at the end of day 1 is 4 milligrams, and it increases 4 times in mass every day. So after 6 days, the mass should be calculated as the initial mass multiplied by 4 raised to the power of the number of days (since it's a daily 4 - fold increase). The initial mass is 4 mg, and after 6 days, the mass ( M = 4\times4\times4\times4\times4\times4=4^{6}) (because we multiply 4 six times, once for each day from day 1 to day 6). Also, we can rewrite (4^{6}) in other equivalent forms. We know that (4 = 2^{2}), but also, we can look at the given expressions:
- (4^{6}): This is the mass since it's 4 multiplied by itself 6 times (once per day for 6 days), so it represents the mass.
- (6^{4}): This is 6 multiplied by itself 4 times, which is not related to the 4 - fold daily increase of the initial 4 mg mass, so it does not represent the mass.
- (4\times6): This is 4 multiplied by 6, which is not the same as 4 multiplied by itself 6 times, so it does not represent the mass.
- (4\times4\times4\times4\times4\times4): This is the same as (4^{6}), so it represents the mass.
- (6\times6\times6\times6): This is (6^{4}), which is not related to the growth of the bacteria mass, so it does not represent the mass.
- (16\times256): First, (4^{2}=16) and (4^{4} = 256), so (16\times256=4^{2}\times4^{4}=4^{2 + 4}=4^{6}), so it represents the mass.
Now we categorize each expression:
Step 1: Analyze (4^{6})
The mass after 6 days is (4\times4\times4\times4\times4\times4 = 4^{6}), so this represents the mass.
Step 2: Analyze (6^{4})
(6^{4}=6\times6\times6\times6), which has nothing to do with the 4 - fold daily growth of the 4 mg initial mass, so it does not represent the mass.
Step 3: Analyze (4\times6)
(4\times6 = 24), while the actual mass after 6 days is (4^{6}=4096), so this does not represent the mass.
Step 4: Analyze (4\times4\times4\times4\times4\times4)
This is the product of 4 six times, which is the formula for the mass after 6 days (initial mass 4, multiplied by 4 each day for 6 days), so it represents the mass.
Step 5: Analyze (6\times6\times6\times6)
This is (6^{4}), not related to the bacteria's mass growth, so it does not represent the mass.
Step 6: Analyze (16\times256)
Since (16 = 4^{2}) and (256=4^{4}), (16\times256 = 4^{2}\times4^{4}=4^{6}), so it represents the mass.
Now we fill the table:
| Represents the mass, in milligrams, of the bacteria sample after 6 days | Does not represent the mass, in milligrams, of the bacteria sample after 6 days |
|---|---|
| (4^{6}), (4\times4\times4\times4\times4\times4), (16\times256) | (6^{4}), (4\times6), (6\times6\times6\times6) |
Answer:
Represents the mass, in milligrams, of the bacteria sample after 6 days:
- (4^{6})
- (4\times4\times4\times4\times4\times4)
- (16\times256)
Does not represent the mass, in milligrams, of the bacteria sample after 6 days:
- (6^{4})
- (4\times6)
- (6\times6\times6\times6)