Problem 42 Let \(f(t)=(t+6)^{2} .\) Find \(... [FREE SOLUTION]

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Chapter 11: Problem 42

Let \(f(t)=(t+6)^{2} .\) Find \(t\) such that \(f(t)=15\).

Short Answer

Expert verified

t = -6 卤鈭�15

Step by step solution

Set up the equation

Start by setting the function equal to 15: \[ (t + 6)^{2} = 15 \]

Take the square root of both sides

To eliminate the square, take the square root of both sides of the equation: \[ t + 6 = \pm \sqrt{15} \]

Solve for t

Isolate t by subtracting 6 from both sides. This gives two possible solutions: \[ t = -6 + \sqrt{15} \] \[ t = -6 - \sqrt{15} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Evaluation

Function evaluation is the process of finding the outputs of a function for specific inputs. In our example, we have a function defined as \(f(t) = (t + 6)^2\).
We are asked to find \(t\) such that \(f(t) = 15\).
To start off, we set \(f(t)\) equal to 15 and get the equation: \[(t + 6)^2 = 15\]
This equation tells us that the square of the term \(t + 6\) must equal 15. Our goal is to find the value(s) of \(t\) that will satisfy this equation.

Square Root Property

The square root property is a powerful tool for solving equations that involve squares. When you have an equation of the form \(x^2 = k\), you can take the square root of both sides to solve for \(x\).
It's important to remember that taking the square root of a number gives two solutions: one positive and one negative.
For example, with \((t + 6)^2 = 15\), applying the square root to both sides gives: \[t + 6 = \pm \sqrt{15}\]
Here, the \(\pm\) symbol represents two possible solutions: \(\sqrt{15}\) and \(-\sqrt{15}\). This is crucial because squaring either one of these will result in the original value 15.

Isolating Variables

Once we have applied the square root property, we must isolate the variable \(t\) to solve the equation completely. Isolating the variable means getting it alone on one side of the equation.
After taking the square root, we have: \[t + 6 = \sqrt{15}\] and \[t + 6 = -\sqrt{15}\]
To isolate \(t\), we need to subtract 6 from both sides of each equation. This gives us two possible solutions: \[t = -6 + \sqrt{15}\] and \[t = -6 - \sqrt{15}\]
These are the two values for \(t\) that satisfy the original function equation \((t+6)^2 = 15\). By isolating the variable, we make sure to find the exact solutions.

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91影视

Let \(f(t)=(t+6)^{2} .\) Find \(t\) such that \(f(t)=15\).

Short Answer

Step by step solution

Set up the equation

Take the square root of both sides

Solve for t

Key Concepts

Function Evaluation

Square Root Property

Isolating Variables

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