Calculus Examples

$x = 9 t^{2}$ and $y = 27 t^{3}$

Step 1

Set up the parametric equation for $x (t)$ to solve the equation for $t$ .

$x = 9 t^{2}$

Step 2

Rewrite the equation as $9 t^{2} = x$ .

$9 t^{2} = x$

Step 3

Divide each term in $9 t^{2} = x$ by $9$ and simplify.

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Step 3.1

Divide each term in $9 t^{2} = x$ by $9$ .

$\frac{9 t^{2}}{9} = \frac{x}{9}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $9$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{9 t^{2}}{9} = \frac{x}{9}$

Step 3.2.1.2

Divide $t^{2}$ by $1$ .

$t^{2} = \frac{x}{9}$

Step 4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$t = \pm \sqrt{\frac{x}{9}}$

Step 5

Simplify $\pm \sqrt{\frac{x}{9}}$ .

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Step 5.1

Rewrite $\frac{x}{9}$ as ${(\frac{1}{3})}^{2} x$ .

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Step 5.1.1

Factor the perfect power $1^{2}$ out of $x$ .

$t = \pm \sqrt{\frac{1^{2} x}{9}}$

Step 5.1.2

Factor the perfect power $3^{2}$ out of $9$ .

$t = \pm \sqrt{\frac{1^{2} x}{3^{2} \cdot 1}}$

Step 5.1.3

Rearrange the fraction $\frac{1^{2} x}{3^{2} \cdot 1}$ .

$t = \pm \sqrt{{(\frac{1}{3})}^{2} x}$

Step 5.2

Pull terms out from under the radical.

$t = \pm \frac{1}{3} \sqrt{x}$

Step 5.3

Combine $\frac{1}{3}$ and $\sqrt{x}$ .

$t = \pm \frac{\sqrt{x}}{3}$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.1

First, use the positive value of the $\pm$ to find the first solution.

$t = \frac{\sqrt{x}}{3}$

Step 6.2

Next, use the negative value of the $\pm$ to find the second solution.

$t = - \frac{\sqrt{x}}{3}$

Step 6.3

The complete solution is the result of both the positive and negative portions of the solution.

$t = \frac{\sqrt{x}}{3}$

$t = - \frac{\sqrt{x}}{3}$

$t = \frac{\sqrt{x}}{3}$

$t = - \frac{\sqrt{x}}{3}$

Step 7

Replace $t$ in the equation for $y$ to get the equation in terms of $x$ .

$y = 27 {(\frac{\sqrt{x}}{3}, - \frac{\sqrt{x}}{3})}^{3}$