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3 years ago

In a 30-60-90 triangle, how many sides do you need to know in order to determine the remaining sides?

Mathematics

2 answers:

yaroslaw [1]3 years ago

7 0

1, because you just need one to determine the remaining sides

butalik [34]3 years ago

4 0

If you know all the angles you only need to know 1 side

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Let S be the surface defined by x 2 + 2y 3 + 3z 4 = 6. Let T be the surface defined parametrically by r(u, v) = (1+ln u, 2e v+u−

aleksandrvk [35]

The tangent to $C$ through (1, 1, 1) must be perpendicular to the normal vectors to the surfaces $S$ and $T$ at that point.

Let $f(x,y,z)=x^2+2y^3+3z^4$ . Then $S$ is the level curve $f(x,y,z)=6$ . Recall that the gradient vector is perpendicular to level curves; we have

$\nabla f(x,y,z)=(2x,6y,12z^2)$

so that the gradient of $f$ at (1, 1, 1) is

$\nabla f(1,1,1)=(2,6,12)$

For the surface $T$ , we have

$\begin{cases}1+\ln u=1\\2e^v+u-2=1\\uv+1=1\end{cases}\implies u=1,v=0$

so that $\vec r(1,0)=(1,1,1)$ . We can obtain a vector normal to $T$ by taking the cross product of the partial derivatives of $\vec r(u,v)$ , and evaluating that product for $u=1,v=0$ :

$\dfrac{\partial\vec r}{\partial u}\times\dfrac{\partial\vec r}{\partial v}=\left(u-2ve^v,-1,\dfrac{2e^v}u\right)$

$\left(\dfrac{\partial\vec r}{\partial u}\times\dfrac{\partial\vec r}{\partial v}\right)(1,0)=(1,-1,2)$

Now take the cross product of the two normal vectors to $S$ and $T$ :

$(2,6,12)\times(1,-1,2)=(24,8,-8)$

The direction of vector (24, 8, -8) is the direction of the tangent line to $C$ at (1, 1, 1). We can capture all points on the line containing this vector by scaling it by $t\in\Bbb R$ . Then adding (1, 1, 1) shifts this line to the point of tangency on $C$ . So the tangent line has equation

$\vec\ell(t)=(1,1,1)+t(24,8,-8)=(1+24t,1+8t,1-8t)$

7 0

3 years ago

0.006 is 1\10 of what

nevsk [136]

Answer: 0.006 is 1/10 of 0.06
Explanation: if you multiply 0.006 by 10 then you get 0.06

5 0

4 years ago

Read 2 more answers

What is 3 to the power 3/7 rounded to the nearest half

Leno4ka [110]

Answer:

1 1/2

Step-by-step explanation:

3^3/7 ≈ 1.601

1.601 rounded to the nearest half is 1 1/2.

3 0

3 years ago

What is the greatest common factor that could be used to reduce 36/90

saw5 [17]

Answer:

Step-by-step explanation:

The gcf is 18

36 divided by 18 is 2

90 divided by 18 is 5

The simplified fraction is 2/5 btw

7 0

3 years ago

Claire and Ashley are soccer players. Throughout the last soccer season, Claire scored three times as many goals as Ashley did.

Artemon [7]

The expression could be (1/3)C.

If Claire scored three times as many as Ashley, that means that Ashley scored 1/3 the number that Claire did.

Simply multiply the amount that Claire score by 1/3 and you have the number that Ashley scored.

8 0

3 years ago