Question

a. the diameter of the sun is approximately 1.391×10⁶ kilometres. the diameter of earth is approximately 12 742 kilometres. approximately how many earths could fit along the diameter of the sun?
b. one water molecule weighs approximately 2.989×10⁻²⁶ kilograms. how many water molecules are there in lake ontario, which contains approximately 1.64×10¹⁵ kilograms of water?
c. diatoms are a group of microalgae found in oceans, waterways, and soil. they are a primary source of food in the sea. one type of diatom measures 0.07 millimetres in length. how many diatoms would be needed to stretch along 1200 kilometres of shoreline?

Explanation:

Step1: Convert Earth's diameter to scientific - notation

The diameter of Earth is $12742 = 1.2742\times10^{4}$ km. To find how many Earths could fit along the diameter of the sun, divide the sun's diameter by Earth's diameter. Let $n_1$ be the number of Earths. Then $n_1=\frac{1.391\times 10^{6}}{1.2742\times 10^{4}}$.
Using the rule $\frac{a\times10^{m}}{b\times10^{n}}=\frac{a}{b}\times10^{m - n}$, we have $n_1=\frac{1.391}{1.2742}\times10^{6 - 4}\approx1.092\times10^{2}=109.2\approx109$.

Step2: Find the number of water - molecules

Let $n_2$ be the number of water molecules in Lake Ontario. Given the weight of one water molecule is $2.989\times 10^{-26}$ kg and the weight of water in Lake Ontario is $1.64\times 10^{15}$ kg. Then $n_2=\frac{1.64\times 10^{15}}{2.989\times 10^{-26}}$.
Using the rule $\frac{a\times10^{m}}{b\times10^{n}}=\frac{a}{b}\times10^{m - n}$, we get $n_2=\frac{1.64}{2.989}\times10^{15-(-26)}\approx0.5487\times10^{41}=5.487\times 10^{40}$.

Step3: Convert units and find the number of diatoms

First, convert 1200 km to mm. Since $1$ km = $10^{6}$ mm, then $1200$ km=$1200\times10^{6}=1.2\times 10^{9}$ mm.
Let $n_3$ be the number of diatoms. The length of one diatom is $0.07$ mm. Then $n_3=\frac{1.2\times 10^{9}}{0.07}=\frac{1.2\times 10^{9}}{7\times 10^{-2}}$.
Using the rule $\frac{a\times10^{m}}{b\times10^{n}}=\frac{a}{b}\times10^{m - n}$, we have $n_3=\frac{1.2}{7}\times10^{9-(-2)}\approx0.1714\times10^{11}=1.714\times 10^{10}$.

Answer:

a. Approximately 109 Earths could fit along the diameter of the sun.
b. There are approximately $5.487\times 10^{40}$ water molecules in Lake Ontario.
c. Approximately $1.714\times 10^{10}$ diatoms would be needed to stretch along 1200 km of shoreline.