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Chapter 2: Problem 113

a. If \(y=k x^{2},\) find the value of \(k\) using \(x=2\) and \(y=64\). b. Substitute the value for \(k\) into \(y=k x^{2}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=5\).

Short Answer

Expert verified
a. The value of \(k\) is 16. b. The new equation is \(y = 16x^{2}\). c. The value of \(y\) when \(x = 5\) is \(400\).

Step by step solution

01

Calculate the value of k

To find the value of \(k\), substitute the given values of \(x = 2\) and \(y = 64\) into the formula \(y = kx^{2}\). Solve for \(k\) by dividing both sides of the equation by \(x^{2}\).
02

Substitute the value of k into the equation

Substitute the value of \(k\) found in step 1 into the original equation \(y = kx^{2}\). This will yield a new equation.
03

Find the value of y

Use the equation obtained in step 2 to find the value of \(y\) when \(x = 5\). To do this, substitute \(x = 5\) into the equation and solve for \(y\).

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Key Concepts

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

Solving for Variables
When solving for variables in a quadratic function like \( y = kx^2 \), we need to determine the value of a specific letter or symbol, known as a "variable." In the given problem, the goal is to find the value of \( k \). A practical way to start is by substituting known values into the equation, like \( x = 2 \) and \( y = 64 \).
Here's how it works:
  • Substitute the given values into the equation: \( 64 = k \times 2^2 \).
  • Calculate \( 2^2 \) which equals \( 4 \).
  • The equation now becomes \( 64 = 4k \).
  • To isolate \( k \), divide both sides by \( 4 \), resulting in \( k = 16 \).
Now that we've solved for \( k \), we can use this value in the next steps.
Substituting Values
Substituting values is crucial for understanding how equations work. Once we have a solution to one part of a problem, like finding \( k \), we carry this forward into new equations. It's a step-by-step transition of information.
Here's how substitution is performed:
  • Take the value of \( k \) we found (\( k = 16 \)) and substitute it back into the original equation.
  • That equation is \( y = kx^2 \). Now it becomes \( y = 16x^2 \).
  • This transformed equation represents a specific quadratic function, which we'll use to find other values.
By substituting \( k = 16 \), we generate a clearer equation that makes further calculations straightforward.
Mathematical Equations
Math equations help represent relationships between numbers and variables clearly. From the newly formed equation \( y = 16x^2 \), we can further explore its implications. This process involves finding unknowns given some conditions.
To find \( y \) for a new value of \( x \):
  • Identify that we need to find \( y \) when \( x = 5 \).
  • Substitute \( x = 5 \) into the equation \( y = 16x^2 \).
  • Calculate \( 5^2 \), which is \( 25 \).
  • Multiply \( 16 \times 25 \) to get \( y = 400 \).
In this problem, mathematical equations are tools that let us see how \( y \) changes with different values of \( x \). This method of solving allows us to accurately predict outcomes under varying circumstances.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. Solve: \(2 x^{2}+x=15\)

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