For what value of $a$ will $8 \times 10^3$ be 40 times larger than $a \times 10^2$?
The problem asks us to find the value of \(a\) such that \(8 \times 10^3\) is equal to 40 times \(a \times 10^2\). We can translate this statement into the following equation:
\[8 \times 10^3 = 40 \times (a \times 10^2)\]
Our goal is to isolate \(a\) and solve for its value.
First, let's simplify both sides of the equation by evaluating the powers of 10.
We know that:
\[10^3 = 1000\]
\[10^2 = 100\]
Substituting these values back into our equation, we get:
\[8 \times 1000 = 40 \times (a \times 100)\]
\[8000 = 40 \times 100 \times a\]
Now, let's perform the multiplication on the right side of the equation:
\[40 \times 100 = 4000\]
So, the equation becomes:
\[8000 = 4000 \times a\]
To solve for \(a\), we divide both sides of the equation by 4000:
\[a = \frac{8000}{4000}\]
\[a = 2\]
Thus, the value of \(a\) is 2.
Scientific notation is commonly used in physics and engineering to represent very large or very small numbers efficiently. For example, if a company has a budget represented as \(\$8 \times 10^3\) (eight thousand dollars) and it is 40 times the budget of a smaller project represented by \(a \times 10^2\), we can use the same algebraic steps as above to determine that the smaller project's budget base is \(a = 2\), or \(\$200\). This helps in scaling and comparing data sets across different magnitudes easily.