If x is greater than or equal to 100, then √x+16 (the whole thing is square rooted) is between
A. √x and √x+1
B. √x+1 and √x+2
C. √x+2 and √x+3
D. √x+3 and √x+4
How do I go about solving this?
To solve this problem, let's go step by step.
First, we need to understand the property of square roots. The square root (√) of a number x is the value that, when multiplied by itself, equals x. So, √x means the number whose square is x.
Now, let's analyze the given expression: √x+16.
To determine its range between two set values, we need to find its lower and upper bounds.
Since x is greater than or equal to 100, we can start by considering x = 100.
Substituting this value into the expression, we have:
√100 + 16 = 10 + 16 = 26.
So, the lower bound of the expression is 26.
Next, let's consider the upper bound. As x increases, the value inside the square root also increases. It means √x increases as well.
Since the options given to us are in the form of √x + a (where a is a constant), we can see that as x increases, the expression √x + 16 also increases.
To find the upper bound, let's consider x = 10000.
Substituting this value into the expression, we have:
√10000 + 16 = 100 + 16 = 116.
So, the upper bound of the expression is 116.
Now, we need to determine between which two numbers (in terms of square roots) the expression √x + 16 falls.
To do this, we can calculate the square roots of x, x + 1, x + 2, and so on until we find the range that includes the expression's values (26 to 116).
For example:
√100 = 10
√101 ≈ 10.05
√102 ≈ 10.1
√103 ≈ 10.15
We keep calculating until we reach a value greater than or equal to 26 and a value less than or equal to 116.
After calculating, we find that:
√100 < √x + 16 < √101
Therefore, the answer is A. √x and √x + 1.
By following this step-by-step process, we were able to determine the solution.