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bogdanovich [222]
3 years ago
11

At what points does the curve r(t) = ti + (5t − t2)k intersect the paraboloid z = x2 + y2? (if an answer does not exist, enter d

ne.)
Mathematics
2 answers:
Sergeu [11.5K]3 years ago
6 0

Answer : curve r(t) = t i + (2t - t 2)k intersect the paraboloid z = x 2 + y 2

By the equation of r(t), x = t, y = 0, and z = 5t - t^{2}.  

Thus, z = x^{2} + y^{2}  

on solving we get,

5t - t^{2} = t^{2} + 0^{2}  

Now, we can solve for t  

t^{2} + t^{2} -5t = 0

So, 2t^2 - 5t = 0

2t ( t - 5/2) = 0  

t = 0 , t = 5  

plugging these values in t = 0 into r(t)

r(0) = <0 , 0 , 0>

r(5) = < 5, 0 , 25 >

Reptile [31]3 years ago
5 0

The curve r\left( t \right) = ti + \left( {5t - {t^2}} \right)k intersect the paraboloid z = {x^2} + {y^2} at \boxed{t = 0} and \boxed{t = \frac{5}{2}}.

Further explanation:

Given:

The equation of the curve r\left( t \right) = ti + \left( {5t - {t^2}} \right)k.

The equation of the paraboloid z = {x^2} + {y^2}.

Explanation:

From theequation of the curve r\left( t \right) = ti + \left( {5t - {t^2}} \right)k.

The value of x is t and the value of y is 0 and the equation of z is z = 5t - {t^2}.

Substitute t for x, 0 for y and 5t - {t^2} in equation of the paraboloid

\begin{aligned}5t - {t^2} &= {t^2} + 0\\0&= {t^2} + {t^2} - 5t\\0&= 2{t^2} - 5t\\0&= 2t\left( {t - 5} \right)\\t&= 0\;{\text{or}}\;t - 5 &= 0\\t &= 0\;{\text{or }}t&= 5\\\end{aligned}

The value of r\left( 0 \right) can be obtained as follows,

\begin{aligned}r\left( 0 \right)&= 0 + \left( {0 - {0^2}} \right)k\\&= 0\\\end{aligned}

The value of y is 0 and the value of z can be obtained as follows,

\begin{aligned}z&= {5^2} + {0^2}\\&=25\\\end{aligned}

The curve r\left( t \right) = ti + \left( {5t - {t^2}} \right)k intersect the paraboloid z = {x^2} + {y^2} at \boxed{t = 0} and \boxed{t = \frac{5}{2}}.

Learn more:

  1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.
  2. Learn more about equation of circle brainly.com/question/1506955.
  3. Learn more about range and domain of the function brainly.com/question/3412497

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Intersecting curves

Keywords: Points, curve, intersects, z=x2+y2,paraboloid, not, exist, r(t) = ti + (5t – t2)k, at what.

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