Question

the width of a grain of yeast is approximately 4 micrometers, which is 4 × 10^(-6) meter. give your answers in scientific notation. part a what would be the width, in meters, of 6.8 × 10^7 grains of yeast laid end to end? part b if a grain of salt is 0.4 mm wide, how many grains of yeast could fit across one grain of salt? explain. (hint: 1 mm = 1 × 10^(-3) m.)

Explanation:

Step1: Multiply number of grains by width of one grain

We know the width of one - grain of yeast is $4\times10^{-6}$ meter and the number of grains is $6.8\times 10^{7}$. We use the formula for total width $W = n\times w$, where $n$ is the number of grains and $w$ is the width of one grain. So $W=(6.8\times 10^{7})\times(4\times 10^{-6})$.

Step2: Use the rules of exponents for multiplication

According to the rule $a^{m}\times a^{n}=a^{m + n}$ and $(a\times b)\times(c\times d)=(a\times c)\times(b\times d)$. So $(6.8\times 10^{7})\times(4\times 10^{-6})=(6.8\times4)\times(10^{7}\times10^{-6})$.
$6.8\times4 = 27.2$ and $10^{7}\times10^{-6}=10^{7+( - 6)}=10^{1}$. Then $27.2\times10^{1}=2.72\times10^{2}$ meters.

Step3: Convert width of salt to meters

The width of a grain of salt is $0.4$ mm. Since $1$ mm=$1\times 10^{-3}$ m, then the width of a grain of salt in meters is $0.4\times10^{-3}=4\times10^{-4}$ m.

Step4: Calculate number of yeast grains across salt grain

Let the number of yeast grains be $N$. We use the formula $N=\frac{\text{width of salt grain}}{\text{width of yeast grain}}$. So $N=\frac{4\times 10^{-4}}{4\times 10^{-6}}$.

Step5: Use the rules of exponents for division

According to the rule $\frac{a^{m}}{a^{n}}=a^{m - n}$ and $\frac{a\times b}{c\times d}=\frac{a}{c}\times\frac{b}{d}$. So $\frac{4\times 10^{-4}}{4\times 10^{-6}}=\frac{4}{4}\times\frac{10^{-4}}{10^{-6}}$.
$\frac{4}{4}=1$ and $\frac{10^{-4}}{10^{-6}}=10^{-4-( - 6)}=10^{2}$. So $N = 100$ grains.

Answer:

Part A: $2.72\times 10^{2}$ meters
Part B: 100 grains. We first convert the width of the salt grain from millimeters to meters ($0.4$ mm = $4\times10^{-4}$ m). Then, by dividing the width of the salt grain by the width of a yeast grain ($4\times10^{-6}$ m), using the rule of exponents for division $\frac{a^{m}}{a^{n}}=a^{m - n}$, we get $\frac{4\times 10^{-4}}{4\times 10^{-6}} = 100$.