Question

In 10 days, the Earth picks up \[2.6 \times {10^8}\] pounds of dust from the atmosphere. How much dust will it pick up in 45 days?

Answer 1

Hint:
Here, we need to find the amount of dust picked up by the Earth in 45 days. First, we will find the amount of dust picked up in 1 day. Then, we will find the amount of dust picked up in 45 days. We will use the rules of exponents to simplify the expressions.
Formula Used: Rules of exponents are \[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\] and \[{a^m} \times {a^n} = {a^{m + n}}\].

Complete step by step solution:
We will use a unitary method and the rules of exponents to get the amount of dust picked up in 45 days.
First, we will calculate the amount of dust the Earth picks up in 1 day.
We know that the Earth picks up \[2.6 \times {10^8}\] pounds of dust from the atmosphere in 10 days.
Therefore, dividing \[2.6 \times {10^8}\] by 10, we get
Amount of dust picked up by the Earth from the atmosphere in 1 day \[ = \dfrac{{2.6 \times {{10}^8}}}{{10}}\]
Rewriting the expression, we get
\[\begin{array}{c} \Rightarrow \dfrac{{2.6 \times {{10}^8}}}{{10}} = \dfrac{{2.6 \times {{10}^8}}}{{{{10}^1}}}\\ \Rightarrow \dfrac{{2.6 \times {{10}^8}}}{{10}} = 2.6 \times \dfrac{{{{10}^8}}}{{{{10}^1}}}\end{array}\]
Now, we will use the rule of exponent \[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\] to simplify the expression.
Thus, we get
\[\begin{array}{l} \Rightarrow \dfrac{{2.6 \times {{10}^8}}}{{10}} = 2.6 \times {10^{8 - 1}}\\ \Rightarrow \dfrac{{2.6 \times {{10}^8}}}{{10}} = 2.6 \times {10^7}\end{array}\]
Therefore, the amount of dust picked up by the Earth from the atmosphere in 1 day \[ = 2.6 \times {10^7}\] pounds.
Next, the amount of dust picked up by the Earth in 45 days is the product of the amount of dust picked up in 1 day multiplied by the number of days.
Multiplying the expression \[2.6 \times {10^7}\] by 45, we get
Amount of dust picked up by the Earth from the atmosphere in 45 days \[ = 2.6 \times {10^7} \times 45\]
Multiplying \[2.6\] and 45 in the expression, we get
Amount of dust picked up by the Earth from the atmosphere in 45 days \[ = 117 \times {10^7}\]
Rewriting the expression, we get
\[\begin{array}{c} \Rightarrow 117 \times {10^7} = 11.7 \times 10 \times {10^7}\\ \Rightarrow 117 \times {10^7} = 11.7 \times {10^1} \times {10^7}\end{array}\]
Now, we will use the rule of exponent \[{a^m} \times {a^n} = {a^{m + n}}\] to simplify the expression.
Thus, we get
\[\begin{array}{c} \Rightarrow 117 \times {10^7} = 11.7 \times {10^{7 + 1}}\\ \Rightarrow 117 \times {10^7} = 11.7 \times {10^8}\end{array}\]

\[\therefore\] The amount of dust picked up by the Earth in 45 days is \[11.7 \times {10^8}\] pounds.

Note:
We used a unitary method to solve the problem. Unitary method is a method where first, the per unit quantity is calculated, and then the number of units are multiplied. Here, we calculated the per day amount of dust picked and multiplied it by 45 to get the amount of dust picked in 45 days.