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Nezavi [6.7K]
3 years ago
11

if the line〈3 + 2t,1 +t,2−t〉intersects the unit sphere inR3given byx2+y2+z2= 1,and if so, at what points.

Mathematics
1 answer:
damaskus [11]3 years ago
6 0

Answer:

( x_1 , y_1 , z_1 ) = < -7 + 4\sqrt{3} , -4 + 2\sqrt{3} , 7 - 2\sqrt{3} >\\\\( x_2 , y_2 , z_2 ) = < -7 - 4\sqrt{3} , -4 - 2\sqrt{3} , 7 + 2\sqrt{3} >\\

Step-by-step explanation:

Solution:-

- We are given a parametric form for the vector equation of line defined by ( t ).

- The line vector equation is:

                        L: < 3 + 2t , t + 1 , 2 -t >

- The same 3-dimensional space is occupied by a unit sphere defined by the following equation:

                          S: x^2 + y^2 + z^2  = 1

- We are to determine the points of intersection of the line ( L ) and the unit sphere ( S ).

- We will substitute the parametric equation of line ( L ) into the equation defining the unit sphere ( S ) and solve for the values of the parameter ( t ):

                        ( 3 + 2t )^2 + ( 1 + t )^2 + ( 2 - t)^2 = 1\\\\( 9 + 12t + 4t^2 ) + ( t^2 + 2t + 1 ) + ( 4 + t^2 -4t ) = 1\\\\t^2 + 10t + 13 = 0\\\\

- Solve the quadratic equation for the parameter ( t ):

                       t = -5 + 2\sqrt{3} , -5 - 2\sqrt{3}

- Plug in each of the parameter value in the given vector equation of line and determine a pair of intersecting coordinates:

                      ( x_1 , y_1 , z_1 ) = < -7 + 4\sqrt{3} , -4 + 2\sqrt{3} , 7 - 2\sqrt{3} >\\\\( x_2 , y_2 , z_2 ) = < -7 - 4\sqrt{3} , -4 - 2\sqrt{3} , 7 + 2\sqrt{3} >\\

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3 years ago
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Anvisha [2.4K]

Answer:

y''(-1) =8

General Formulas and Concepts:

<u>Pre-Algebra</u>

Order of Operations: BPEMDAS

  1. Brackets
  2. Parenthesis
  3. Exponents
  4. Multiplication
  5. Division
  6. Addition
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  • Left to Right

Equality Properties

<u>Algebra I</u>

  • Factoring

<u>Calculus</u>

Implicit Differentiation

The derivative of a constant is equal to 0

Basic Power Rule:

  • f(x) = cxⁿ
  • f’(x) = c·nxⁿ⁻¹

Product Rule: \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)

Chain Rule: \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)

Quotient Rule: \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}

Step-by-step explanation:

<u>Step 1: Define</u>

-xy - 2y = -4

Rate of change of the tangent line at point (-1, 4)

<u>Step 2: Differentiate Pt. 1</u>

<em>Find 1st Derivative</em>

  1. Implicit Differentiation [Product Rule/Basic Power Rule]:                            -y - xy' - 2y' = 0
  2. [Algebra] Isolate <em>y'</em> terms:                                                                               -xy' - 2y' = y
  3. [Algebra] Factor <em>y'</em>:                                                                                       y'(-x - 2) = y
  4. [Algebra] Isolate <em>y'</em>:                                                                                         y' = \frac{y}{-x-2}
  5. [Algebra] Rewrite:                                                                                           y' = \frac{-y}{x+2}

<u>Step 3: Find </u><em><u>y</u></em>

  1. Define equation:                    -xy - 2y = -4
  2. Factor <em>y</em>:                                 y(-x - 2) = -4
  3. Isolate <em>y</em>:                                 y = \frac{-4}{-x-2}
  4. Simplify:                                 y = \frac{4}{x+2}

<u>Step 4: Rewrite 1st Derivative</u>

  1. [Algebra] Substitute in <em>y</em>:                                                                               y' = \frac{-\frac{4}{x+2} }{x+2}
  2. [Algebra] Simplify:                                                                                         y' = \frac{-4}{(x+2)^2}

<u>Step 5: Differentiate Pt. 2</u>

<em>Find 2nd Derivative</em>

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<u>Step 6: Find Slope at Given Point</u>

  1. [Algebra] Substitute in <em>x</em>:                                                                               y''(-1) = \frac{8}{(-1+2)^3}
  2. [Algebra] Evaluate:                                                                                       y''(-1) =8
6 0
3 years ago
Read 2 more answers
A. A fair 6-face die will be rolled some number of times, and you win if it shows an ace (one spot) 20% or more of the time. Oth
butalik [34]

Answer: The complete question is found in the attachment

Step-by-step explanation:

Law of large numbers: The probability of occurrence of an event becomes closer to the theoretical probability as the number of trials increases

P(an ace) = 1/6

= 0.1667

= 16.67%

a) In 600 rolls, the value will be close to 16.67. compared to 60 rolls

Greater than 20% interval doesn't include 16.67%. So, for more than 20% ace, 60 rolls is better.

b) More than 15% interval includes 16.67. So, it is better to roll 600 times

c) The interval between 15% and 20% include 16.67% and hence, 600 rolls is better

d) Larger number of trials is better to get exactly 16

So, 600 rolls is better

4 0
3 years ago
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nika2105 [10]

Answer:

4. -4

5. -2

6. -3

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Divide both sides by -6

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6. -14 = 4x - 2

Add 2 to both sides;

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Divide both sides by 4;

-3 = x OR x = -3

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