Question Statement

A particle moves on the graph of $y^2=x+1$ such that the rate of change of $x$ with respect to time is given by $\frac{dx}{dt}=4x+4$. What is $\frac{dx}{dt}$ when $x=8$?

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Background and Explanation

This problem requires evaluating a given rate equation at a specific value of $x$. Although the particle's path is defined by the curve $y^2=x+1$, we are provided with an explicit expression for $\frac{dx}{dt}$ as a function of $x$, allowing us to solve by direct substitution without needing to analyze the curve's geometry.

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Solution

We are given the differential equation describing the rate of change of $x$ with respect to time:

$\frac{dx}{dt}=4x+4$

To find the value of $\frac{dx}{dt}$ when the particle is at position $x=8$, we substitute $x=8$ directly into equation (1):

$\begin{array}{rl}\frac{dx}{dt}& =4(8)+4\\ & =32+4\\ & =36\end{array}$

Therefore, when $x=8$, the rate of change $\frac{dx}{dt}$ equals $36$.

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Key Formulas or Methods Used

• Given rate equation: $\frac{dx}{dt}=4x+4$
• Direct substitution: Evaluating a function by replacing the variable with a specific numeric value
• Algebraic order of operations: Multiplication before addition ($4\times 8=32$, then $32+4=36$)

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Summary of Steps

1. Identify the given rate equation: $\frac{dx}{dt}=4x+4$
2. Substitute $x=8$ into the right-hand side of the equation
3. Calculate $4(8)=32$
4. Add $4$ to obtain the final result: $32+4=36$