Find the value of 64 ÷ 2 ∙ 16

Find the unit tangent vector T and the principal unit normal vector N for the following parameterized curve.

const2013 [10]

Answer:

a.

$T(t) = ( -sin(t^2), cos(t^2) )\\\\N(t) = T'(t) / |T'(t) | = (-cos(t^2) , -sin(t^2))$

b.

$T(t) =r'(t)/|r'(t)| = (2t/ \sqrt{4t^2 + 36} , -6/\sqrt{4t^2 + 36},0)$

$N(t) =T(t)/|T'(t)| = (3/(9 + t^2)^{1/2} , t/(9 + t^2)^{1/2},0)$

Step-by-step explanation:

Remember that for any curve $r(t)$

The tangent vector is given by

$T(t) = \frac{r'(t) }{| r'(t)| }$

And the normal vector is given by

$N(t) = \frac{T'(t)}{|T'(t)|}$

a.

For this case, using the chain rule

$r'(t) = ( -10*2tsin(t^2) , 102t cos(t^2) )\\$

And also remember that

$|r'(t)| = \sqrt{(-10*2tsin(t^2))^2 + ( 10*2t cos(t^2) )^2} \\\\ = \sqrt{400 t^2*( sin(t^2)^2 + cos(t^2) ^2 })\\=\sqrt{400t^2} = 20t$

Therefore

$T(t) = r'(t) / |r'(t) | = ( -10*2tsin(t^2) , 10*2t cos(t^2) )/ 20t\\\\ = ( -10*2tsin(t^2)/ 20t , 10*2t cos(t^2) / 20t )\\= ( -sin(t^2), cos(t^2) )$

Similarly, using the quotient rule and the chain rule

$T'(t) = ( -2t cos(t^2) , -2t sin(t^2))$

And also

$|T'(t)| = \sqrt{ ( -2t cos(t^2))^2 + (-2t sin(t^2))^2} = \sqrt{ 4t^2 ( ( cos(t^2))^2 + ( sin(t^2))^2)} = \sqrt{4t^2} \\ = 2t$

Therefore

$N(t) = T'(t) / |T'(t) | = (-cos(t^2) , -sin(t^2))$

Notice that

1. $|N(t)| = |T(t) | = \sqrt{ cos(t^2)^2 + sin(t^2)^2 } = \sqrt{1} = 1$

2. $N(t)*T(T) = cos(t^2) sin(t^2 ) - cos(t^2) sin(t^2 ) = 0$

b.

Simlarly

$r'(t) = (2t,-6,0) \\$

and

$|r'(t)| = \sqrt{(2t)^2 + 6^2} = \sqrt{4t^2 + 36}$

Therefore

$T(t) =r'(t)/|r'(t)| = (2t/ \sqrt{4t^2 + 36} , -6/\sqrt{4t^2 + 36},0)$

Then

$T'(t) = (9/(9 + t^2)^{3/2} , (3 t)/(9 + t^2)^{3/2},0)$

and also

$|T'(t)| = \sqrt{ ( (9/(9 + t^2)^{3/2} )^2 + ( (3 t)/(9 + t^2)^{3/2})^2 + 0^2 }\\= 3/(t^2 + 9 )$

And since

$N(t) =T(t)/|T'(t)| = (3/(9 + t^2)^{1/2} , t/(9 + t^2)^{1/2},0)$

6 0

3 years ago

4) What is the total cost (tax and price) on a purchase of $108.50 if the sales tax is 8%?

vitfil [10]

Answer: C)$117.18

$108.50 X 1.08 = $108.50

4 0

3 years ago

Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.

yan [13]

Answer:

138.16 in²

Step-by-step explanation:

The surface area SA of the cone is the sum of the base area B and the lateral area LA. The lateral area is half the product of the circumference and the slant height. The radius is half the diameter, so is 4 inches.

SA = B + LA

= πr² + (1/2)(2πrh) . . . . where h is the slant height

= (πr)(r +h)

Filling in the numbers, you have ...

SA = (3.14)(4 in)(4 in + 7 in) = 3.14×(44 in²) = 138.16 in²

7 0

3 years ago

In the graph of An inequality, the area above a solid line through the point (-5,2) and (3,2) is shaded. Which inequality does t

allochka39001 [22]

The inequality shown by this graph is y ≥ 2.

<h3>Describe inequality.</h3>

the relationship between two non-equal expressions using a symbol like "not equal to," ≠ "greater than >," or "less than < "

The inequality sign : ≥ or ≤ to use is: or because the line is solid and not dotted.

Given that the points have y-coordinate values of 2, (-5, 2), and (3, 2), respectively, and that the line crosses the y-axis at this point, the proper inequality is:

y ≥ 2

Therefore, y ≥ 2 is the inequality.

To learn more about inequality from the given link:

brainly.com/question/18698351

#SPJ10

8 0

1 year ago

This id hard please help me do this

svlad2 [7]

In this problem, we're going to use the 'PEMDAS' method.

First, we're doing the problem in the parenthesis.
44 - 4 = 40

Next in PEMDAS, we need to do the exponent.
6² = 6 x 6 = 36

Then we would do multiplication, but there is nothing to multiply so we move onto division.
40 / 2 = 20

Now we would add, but there is nothing to add so we move onto subtraction.
20 - 36 = -16

Our final answer would be -16.

Hope this helps!~

7 0

3 years ago

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