Problem 24 - Algebra 1

Question

Evaluate the function for the given value of \(x .\) Round your answer to the nearest tenth. $$y=6 \sqrt{15-x} ;-1$$

Step-by-Step Solution

Answer

The function evaluates to 24 at x=-1.

Step 1: Substitution

The value for x is given as -1. Substitute -1 in place of x in the equation. This will give: \(y=6 \sqrt{15-(-1)}\)

Step 2: Simplifying the inside of the square root

The expression within the square root simplifies to \(15 - (-1) = 16\). The equation then becomes \(y=6 \sqrt{16}\)

Step 3: Solve the square root

The square root of 16 is 4. Hence, the equation now is \(y=6 \cdot 4\)

Step 4: Final calculation

Multiply 6 and 4 to get the value of y as 24.

Key Concepts

Substitution

In mathematics, substitution is a fundamental concept where one replaces a variable with a specific value. Here, the function provided is \( y = 6 \sqrt{15-x} \). To evaluate this function, you'll need to substitute \( x = -1 \) into the equation. Essentially, substitution helps us tailor a general expression to a specific scenario. This allows us to calculate an answer based on given data. Whenever you're working with formulas or equations and you know one of the variable's values, substitution is your go-to strategy. Simply replace every occurrence of the variable with the given number. In this exercise, you replace \( x \) with \(-1\), which transforms the equation into:- \( y = 6 \sqrt{15 - (-1)} \).By inserting \(-1\) in place of \( x \), you prepare the function for further simplification.

Square Root

The square root operation is a method of finding a number which, when multiplied by itself, gives the original number. In our case, the expression within the square root simplifies to \( 15 - (-1) \).The square root is denoted by the radical symbol \( \sqrt{} \), and it is a key element in calculus, algebra, and other areas of mathematics. Here’s the breakdown:- Calculate \( 15 - (-1) \). Applying simple arithmetic shows that subtracting a negative is the same as adding the positive equivalent.This results in \( 15 + 1 = 16 \).This simplifies the equation to \( y = 6 \sqrt{16} \).Now, calculate the square root of 16. Knowing that 4 multiplied by itself yields 16, we identify \( \sqrt{16} = 4 \). Understanding how to compute square roots is crucial since they frequently appear in mathematical equations and real-life applications.

Simplifying Expressions

Simplifying expressions involves reducing complexities to make calculations easier, particularly when dealing with operations like multiplication and division. After identifying that \( \sqrt{16} = 4 \), the expression simplifies further.Guidelines for simplifying expressions include:- Combine like terms.- Execute arithmetic operations following the order of operations (PEMDAS/BODMAS).This entails handling parentheses and radicals first before tackling other operations like multiplication.For our expression, we multiply \( 6 \cdot 4 \), leading to a simple arithmetic calculation.Ultimately, \( 6 \times 4 = 24 \), which is the evaluated function's output. Simplifying helps reach the answer efficiently, transforming a potentially complex process into one that is straightforward.