1986 AJHSME Problems/Problem 7
Problem
How many whole numbers are between and ?
Solution
No... of course you're not supposed to know what the square root of 8 is, or the square root of 80. There aren't any formulas, either. Approximation seems like the best strategy.
Clearly it must be true that for any positive integers , , and with ,
If we let , , and , then we get
Therefore, the smallest whole number between and is .
Similarly, if we let , , and , we get
So is the largest whole number between and .
So we know that we just have to find the number of integers from 3 to 8 inclusive. If we subtract 2 from every number in this set (which doesn't change the number of integers in the set at all), we find that now all we need to do is find the number of integers there are from 1 to 6, which is obviously 6.
Linear Approximation using Calculus
While this may not be the best, most efficient way to solve this problem, it can prove to be quite useful in similar problems. Using Calculus, we can use tangent-line approximation to approximate these square roots. First, we understand that these roots are of the form Given the values to approximate, 8, and 80, we find the closest square number to them, namely, 9, and 81.
Now, we take the derivative of to get We can approximate , to get where is the perfect square closest to the the input value (in this case 8), and is the difference between and the input value.
The two cases are as follows:
Now, we can plug in our values into the equation, getting:
Thus, we see that our answer lies in the amount of whole numbers between and and as we are counting whole numbers, we can round each expression to narrow our interval to Therefore, we see that our answer is
Note: this solution is definitely not the fastest solution, and in a competition, you should probably use intuition to see the closest numbers in the interval of those square roots, and solve the problem from there. ~Cheetahboy93
See Also
1986 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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All AJHSME/AMC 8 Problems and Solutions |
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