Question

Show how to make a quick estimate (to two decimal places) of \(\sqrt{(4.98)^{2}-(3.03)^{2}}\) without using a computer or a calculator. Hint: Consider \(f(x, y)=\sqrt{x^{2}-y^{2}}.\)

Short answer

The quick estimate is approximately 4.00.

Step-by-step solution

Identify Approximations

Approximating the values given: Notice that 4.98 is close to 5 and 3.03 is close to 3. So, use x ≈ 5 and y ≈ 3.

Set Up the Function

Use the function hint provided: \[ f(x, y) = \sqrt{x^2 - y^2} \] Substitute x = 5 and y = 3 into the function.

Apply the Function

Calculate the values: \[ f(5, 3) = \sqrt{5^{2} - 3^{2}} \] This simplifies to \sqrt{25 - 9} = \sqrt{16}.

Simplify the Result

Simplify the resulting square root: \[ \sqrt{16} = 4. \]

Key concepts

Approximation Methods

Approximation methods are useful techniques to quickly estimate the value of complex mathematical expressions without the use of a calculator or computer. These methods rely on substituting values with those that are easier to work with but are close enough to the original ones to give an accurate result.
In our exercise, we approximate 4.98 as 5 and 3.03 as 3. These values are chosen because 5 and 3 are simple to work with and are very close to the original numbers. By doing this, we reduce the complexity of our calculations while keeping the result reasonably accurate.
This method is particularly handy when dealing with functions like square roots, which can be difficult to calculate without a digital tool. Approximation ensures that we can still proceed with our problem-solving using straightforward arithmetic.
Remember, the closer your approximate values are to the original ones, the more accurate your final result will be. However, for many practical purposes, slight deviations are acceptable.

Square Root Function

The square root function, denoted as \(\backslash sqrt{x}\), is a mathematical operation that returns the number which, when multiplied by itself, yields the original number x. For example, \(\backslash sqrt{16} = 4\), because 4 * 4 equals 16.
In this exercise, the function hint provided is \(\backslash sqrt{x^2 - y^2}\). This is an extension of the basic square root function, where we need to first calculate the difference between the squares of two numbers, x and y, before finding their square root.
By using the approximated values for x and y, we simplified our function to \(\backslash sqrt{5^2 - 3^2}\), which further reduces to \(\backslash sqrt{25 - 9}\). This simplifies down to \(\backslash sqrt{16}\), and finally gives us the answer 4.
Understanding how the square root function works and how to manipulate it is key to solving many algebraic problems, especially in cases where direct computation is impractical.

Simplification

Simplification is the process of reducing a mathematical expression to its simplest form. This makes it easier to understand and work with. Simplification often involves combining like terms, performing basic arithmetic operations, and canceling out common factors.
In our exercise, simplification occurs after we substitute the approximated values into the function. We start with \(\backslash sqrt{5^2 - 3^2}\), which translates to \(\backslash sqrt{25 - 9}\). By performing the calculation inside the square root, we reduce the expression to \(\backslash sqrt{16}\).
Finally, by taking the square root of 16, we obtain 4. Each step of simplification makes the problem more manageable and brings us closer to the answer.
Simplification is an essential skill in mathematics as it allows for more efficient problem-solving and clear communication of solutions. By mastering simplification techniques, students can handle more complex problems with confidence.