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A curve has equation $y=\frac{1}{60}(3x+1)^{2}$ and a point is moving along the curve. Find the $x$ -coordinate of the point on the curve at which the $x$ - and $y$ -coordinates are increasing at the same rate. $\quad$

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Find the roots of factored polynomials

Full solution

Q. A curve has equation

y=\frac{1}{60}(3x+1)^{2}

and a point is moving along the curve. Find the

x

-coordinate of the point on the curve at which the

x

- and

y

-coordinates are increasing at the same rate.

\quad

Find Derivative of y: To find where the $x$ and $y$ coordinates are increasing at the same rate, we need to find the derivative of $y$ with respect to $x$ , which will give us the rate of change of $y$ . $\frac{dy}{dx} = \left(\frac{1}{60}\right) \cdot 2 \cdot (3x + 1) \cdot (3)$
Simplify Derivative: Simplify the derivative. $\newline$ $\frac{dy}{dx} = \frac{1}{30} \times (3x + 1) \times 3$ $\newline$ $\frac{dy}{dx} = \frac{1}{10} \times (3x + 1)$
Set Derivative Equal: Set the derivative equal to $1$ since the rate of change of $y$ must equal the rate of change of $x$ , which is $1$ . $\frac{1}{10} \cdot (3x + 1) = 1$
Solve for x: Solve for x. $\newline$ $3x + 1 = 10$ $\newline$ $3x = 9$ $\newline$ $x = 3$

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