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Calculus 2 : Parametric, Polar, and Vector

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Example Questions

Example Question #41 : Parametric, Polar, And Vector

Given x=3t+7 and y=7t+3 , what is in terms (rectangular form)?

Possible Answers:

$y=\frac{7}{3}(x+7)+3$

$y=\frac{3}{7}(x-7)+3$

$y=\frac{7}{3}(x-7)+3$

$y=\frac{3}{7}(x+7)+3$

$y=\frac{7}{3}(x-7)-3$

Correct answer:

$y=\frac{7}{3}(x-7)+3$

Explanation:

Given x=3t+7 and y=7t+3 , let's solve both equations for :

$x=3t+7\rightarrow t=\frac{x-7}{3}$

$y=7t+3\rightarrow t=\frac{y-3}{7}$

Since both equations equal , let's set them equal to each other and solve for :

$\frac{y-3}{7}=\frac{x-7}{3}$

$y-3=7(\frac{x-7}{3})$

$y=\frac{7}{3}(x-7)+3$

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Example Question #42 : Parametric, Polar, And Vector

Given x=7t+5 and y=11t+2 , what is in terms of (rectangular form)?

Possible Answers:

$y=\frac{11}{7}(x+5)+2$

$y=\frac{11}{7}(x+5)-2$

$y=\frac{11}{7}(x-5)+2$

None of the above

$y=\frac{11}{7}(x-5)-2$

Correct answer:

$y=\frac{11}{7}(x-5)+2$

Explanation:

Given x=7t+5 and y=11t+2 , let's solve both equations for :

$x=7t+5\rightarrow t=\frac{x-5}{7}$

$y=11t+2\rightarrow t=\frac{y-2}{11}$

Since both equations equal , let's set them equal to each other and solve for :

$\frac{y-2}{11}=\frac{x-5}{7}$

$y-2=11(\frac{x-5}{7})$

$y=\frac{11}{7}(x-5)+2$

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Example Question #43 : Parametric

Given x=-6t+11 and y=2t-10 , what is in terms of (rectangular form)?

Possible Answers:

$y=\frac{x-11}{3}+10$

$y=\frac{x-11}{3}-10$

$y=-\frac{x-11}{3}+10$

None of the above

$y=-\frac{x-11}{3}-10$

Correct answer:

$y=-\frac{x-11}{3}-10$

Explanation:

Given x=-6t+11 and y=2t-10 let's solve both equations for :

$x=-6t+11\rightarrow t=-\frac{x-11}{6}$

$y=2t-10\rightarrow t=\frac{y+10}{2}$

Since both equations equal , let's set them equal to each other and solve for :

$\frac{y+10}{2}=-\frac{x-11}{6}$

$y+10=-2(\frac{x-11}{6})$

$y=-\frac{x-11}{3}-10$

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Example Question #41 : Parametric, Polar, And Vector

Convert the following equation from parametric to rectangular coordinates:

x=2t^2, y=10t

Possible Answers:

$y=10\sqrt{\frac{x}{2}}$

$y=-10\sqrt{\frac{x}{2}}$

y=5x

$y=\pm10\sqrt{\frac{x}{2}}$

Correct answer:

$y=\pm10\sqrt{\frac{x}{2}}$

Explanation:

To convert the equation from parametric form to rectangular form, we must eliminate the parameter. To do this, we can solve for t with respect to x:

$t=\pm \sqrt{\frac{x}{2}}$

The plus or minus is important to remember because a square root was taken.

Now, simply plug this into the equation for y:

$y=\pm10\sqrt{\frac{x}{2}}$

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Example Question #45 : Parametric

Convert the following to rectangular form from parametric form:

x=2t^3+5 , y=t

Possible Answers:

y=x

$y=\frac{x-5}{2}$

$y=\sqrt[3]{\left(\frac{x-5}{2}\right)}$

$y=\left(\frac{x-5}{2}\right)^{\frac{3}{2}}$

Correct answer:

$y=\sqrt[3]{\left(\frac{x-5}{2}\right)}$

Explanation:

To convert a parametric equation into a rectangular equation, we must eliminate the parameter. We were already given an equation where t was in terms of just a variable:

t=y

Next, substitute this into the equation for x which contains t:

x=2(y^3)+5

Finally, rearrage and solve for y:

$y=\sqrt[3]{\left(\frac{x-5}{2}\right)}$

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Example Question #42 : Parametric, Polar, And Vector

Convert the following equation into rectanglar form:

$x=\cos(t), y=cos^2 (t)+3$

Possible Answers:

$y=\arccos(x)+3$

y=-x^2+3

y=x^2+3

$y=\cos(x)+3$

Correct answer:

y=x^2+3

Explanation:

To convert the given parametric equation into rectangualr coordinates, we must eliminate the parameter by solving for t:

$t=\arccos (x)$

Now replace t with the new term in the equation for y:

$y= (\cos(\textup{arccos}(x)^2)+3=x^2+3$

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Example Question #43 : Parametric, Polar, And Vector

Convert the following equation from parametric to rectangular form:

$x=e^t, y=2\sin(t)$

Possible Answers:

$y=-2\sin(\ln(x))$

y=x

$y=2\sin(e^t)$

$y=2\sin(\ln(x))$

Correct answer:

$y=2\sin(\ln(x))$

Explanation:

To convert from parametric to rectangular form, eliminate the parameter (t) from one of the equations:

$t=\ln(x)$

Now plug this into the equation for y to get our final answer:

$y=2\sin(\ln(x))$

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Example Question #46 : Parametric, Polar, And Vector

Given x=3t-6 and y=7t+9 , what is in terms of (rectangular form)?

Possible Answers:

$y=23x+\frac{7}{3}$

$y=\frac{7}{3}x-23$

$y=23x-\frac{7}{3}$

$y=\frac{7}{3}x+23$

None of the above

Correct answer:

$y=\frac{7}{3}x+23$

Explanation:

Given x=3t-6 and y=7t+9 , let's solve both equations for :

$x=3t-6\rightarrow t=\frac{x+6}{3}$

$y=7t+9\rightarrow t=\frac{y-9}{7}$

Since both equations equal , let's set them equal to each other and solve for :

$\frac{y-9}{7}=\frac{x+6}{3}$

$y-9=\frac{7}{3}(x+6)$

$y-9=\frac{7}{3}x+14$

$y=\frac{7}{3}x+23$

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Example Question #47 : Parametric, Polar, And Vector

Given x=2t+11 and y=11t+2 , what is in terms of (rectangular form)?

Possible Answers:

$y=\frac{11}{2}(x-11)+2$

$y=\frac{11}{2}(x-11)-2$

$y=\frac{11}{2}(x+11)+2$

None of the above

$y=\frac{11}{2}(x+11)-2$

Correct answer:

$y=\frac{11}{2}(x-11)+2$

Explanation:

Given x=2t+11 and y=11t+2 , let's solve both equations for :

$x=2t+11\rightarrow t=\frac{x-11}{2}$

$y=11t+2\rightarrow t=\frac{y-2}{11}$

Since both equations equal , let's set them equal to each other and solve for :

$\frac{y-2}{11}=\frac{x-11}{2}$

$y-2=\frac{11}{2}(x-11)$

$y=\frac{11}{2}(x-11)+2$

Report an Error

Example Question #48 : Parametric, Polar, And Vector

Given x=t+3 and $y=\frac{1}{2}t-3$ , what is in terms of (rectangular form)?

Possible Answers:

$y=\frac{x-3}{2}-3$

None of the above

$y=\frac{x-3}{2}+3$

$y=\frac{x+3}{2}+3$

$y=\frac{x+3}{2}-3$

Correct answer:

$y=\frac{x-3}{2}-3$

Explanation:

Given x=t+3 and $y=\frac{1}{2}t-3$ , let's solve both equations for :

$x=t+3\rightarrow t=x-3$

$y=\frac{1}{2}t-3\rightarrow t=2(y+3)$

Since both equations equal , let's set them equal to each other and solve for :

2(y+3)=x-3

$y+3=\frac{x-3}{2}$

$y=\frac{x-3}{2}-3$

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Calculus 2 : Parametric, Polar, and Vector

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #41 : Parametric, Polar, And Vector

Example Question #42 : Parametric, Polar, And Vector

Example Question #43 : Parametric

Example Question #41 : Parametric, Polar, And Vector

Example Question #45 : Parametric

Example Question #42 : Parametric, Polar, And Vector

Example Question #43 : Parametric, Polar, And Vector

Example Question #46 : Parametric, Polar, And Vector

Example Question #47 : Parametric, Polar, And Vector

Example Question #48 : Parametric, Polar, And Vector

All Calculus 2 Resources

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