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3 years ago

HELPPPPPPPPPP stuckkk

Mathematics

1 answer:

Alenkinab [10]3 years ago

5 0

I think it’s B :):):)):

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Find an equation of the tangent plane to the given parametric surface at the specified point.

Neko [114]

Answer:

Equation of tangent plane to given parametric equation is:

$\frac{\sqrt{3}}{2}x-\frac{1}{2}y+z=\frac{\pi}{3}$

Step-by-step explanation:

Given equation

$r(u, v)=u cos (v)\hat{i}+u sin (v)\hat{j}+v\hat{k}$ ---(1)

Normal vector tangent to plane is:

$\hat{n} = \hat{r_{u}} \times \hat{r_{v}}\\r_{u}=\frac{\partial r}{\partial u}\\r_{v}=\frac{\partial r}{\partial v}$

$\frac{\partial r}{\partial u} =cos(v)\hat{i}+sin(v)\hat{j}\\\frac{\partial r}{\partial v}=-usin(v)\hat{i}+u cos(v)\hat{j}+\hat{k}$

Normal vector tangent to plane is given by:

$r_{u} \times r_{v} =det\left[\begin{array}{ccc}\hat{i}&\hat{j}&\hat{k}\\cos(v)&sin(v)&0\\-usin(v)&ucos(v)&1\end{array}\right]$

Expanding with first row

$\hat{n} = \hat{i} \begin{vmatrix} sin(v)&0\\ucos(v) &1\end{vmatrix}- \hat{j} \begin{vmatrix} cos(v)&0\\-usin(v) &1\end{vmatrix}+\hat{k} \begin{vmatrix} cos(v)&sin(v)\\-usin(v) &ucos(v)\end{vmatrix}\\\hat{n}=sin(v)\hat{i}-cos(v)\hat{j}+u(cos^{2}v+sin^{2}v)\hat{k}\\\hat{n}=sin(v)\hat{i}-cos(v)\hat{j}+u\hat{k}\\$

at u=5, v =π/3

$=\frac{\sqrt{3} }{2}\hat{i}-\frac{1}{2}\hat{j}+\hat{k}$ ---(2)

at u=5, v =π/3 (1) becomes,

$r(5, \frac{\pi}{3})=5 cos (\frac{\pi}{3})\hat{i}+5sin (\frac{\pi}{3})\hat{j}+\frac{\pi}{3}\hat{k}$

$r(5, \frac{\pi}{3})=5(\frac{1}{2})\hat{i}+5 (\frac{\sqrt{3}}{2})\hat{j}+\frac{\pi}{3}\hat{k}$

$r(5, \frac{\pi}{3})=\frac{5}{2}\hat{i}+(\frac{5\sqrt{3}}{2})\hat{j}+\frac{\pi}{3}\hat{k}$

From above eq coordinates of r₀ can be found as:

$r_{o}=(\frac{5}{2},\frac{5\sqrt{3}}{2},\frac{\pi}{3})$

From (2) coordinates of normal vector can be found as

$n=(\frac{\sqrt{3} }{2},-\frac{1}{2},1)$

Equation of tangent line can be found as:

$(\hat{r}-\hat{r_{o}}).\hat{n}=0\\((x-\frac{5}{2})\hat{i}+(y-\frac{5\sqrt{3}}{2})\hat{j}+(z-\frac{\pi}{3})\hat{k})(\frac{\sqrt{3} }{2}\hat{i}-\frac{1}{2}\hat{j}+\hat{k})=0\\\frac{\sqrt{3}}{2}x-\frac{5\sqrt{3}}{4}-\frac{1}{2}y+\frac{5\sqrt{3}}{4}+z-\frac{\pi}{3}=0\\\frac{\sqrt{3}}{2}x-\frac{1}{2}y+z=\frac{\pi}{3}$

5 0

2 years ago

What is the code? Thank you! Whoever answers this, I will give brainliest!

shepuryov [24]

Answer:

5821

Step-by-step explanation:

For the first symbol, the number is 5.

In the first problem, the digit corresponding to that symbol is 5.

In the second one, it should be 5. The area of a triangle is base * height / 2. The base is 15 and the height is 9. Therefore, the area will be 15 * 9 / 2 = 67.5 square units. Now, the number corresponding to that symbol is 5.

In the fourth problem, the number corresponding to the symbol is also 5.

For the second symbol, the number is 8.

In the 5th problem, the number corresponding to that symbol is 8.

In the 7th problem, the number corresponding to the symbol is 8.

In the last problem, 96 is the correct answer, and everything is in the right place. I assume this must be the mistake of the maker of the problem.

For the third symbol, the number is 2

In the 1st and 3rd problems, the number corresponding to that symbol is 2

Finally, for the last symbol, the number is 1

In the 4th problem, there is a trapezoid. The formula for the area of a trapezoid is base 1 * base 2 / 2 * height. base 1 = 29, base 2 = 13. 13 + 29 = 42. 42 / 2 = 21. 21 * 15 = 315. Now, the 2nd digit, the one corresponding to the symbol, is 1.

In the last problem, you need to find the surface area of a figure. I would do this by splitting it up. The area for a triangle is base * height / 2. That means, the area of the triangle would be 8 * 7 / 2 = 28. There are four identical triangles, so you multiply this by 4 and get 112. Next, you find the area of the square which is 7 * 7. 7 * 7 = 49. Now, you add them together. 112 + 49 = 161. Now, the first and last digits corresponding to the symbol are both 1.

The answer is 5821.

4 0

3 years ago

If (a, –5) is a solution to the equation 3a = –2b – 7, what is a?

Llana [10]

Answer:

d) 1⃣

Step-by-step explanation:

Multiply -5 by -2 to get , then deduct 7 to get 3. So, you now know that 3 = 3a; 1 = a.

7 0

3 years ago

As a retiring employee, you'll receive 8.96% of your average salary from the last five years you worked with your company.