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2 years ago

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The isosceles triangle theorem says \"if two sides of a triangle are congruent, then the angles opposite those sides are congrue

nt\". . . If you are using this figure to prove the isosceles triangle theorem, what would be the best strategy?

1 answer:

olga nikolaevna [1]2 years ago

4 0

Draw qs so that s is the midpoint of pr the prove pqs rqs using sss

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A soup recipe calls for 3 chopped onions. You want to make 1/5 of the recipe. How many onions will you need? Which proportion wi

Leya [2.2K]

Answer:

3/5, 1 onion,

Step-by-step explanation

1/5 x 3

8 0

2 years ago

G verify that the divergence theorem is true for the vector field f on the region

Alenkasestr [34]

$\mathbf f(x,y,z)=\langle z,y,x\rangle\implies\nabla\cdot\mathbf f=\dfrac{\partial z}{\partial x}+\dfrac{\partial y}{\partial y}+\dfrac{\partial x}{\partial z}=0+1+0=1$

Converting to spherical coordinates, we have

$\displaystyle\iiint_E\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV=\int_{\varphi=0}^{\varphi=\pi}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=0}^{\rho=6}\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=288\pi$

On the other hand, we can parameterize the boundary of $E$ by

$\mathbf s(u,v)=\langle6\cos u\sin v,6\sin u\sin v,6\cos v\rangle$

with $0\le u\le2\pi$ and $0\le v\le\pi$ . Now, consider the surface element

$\mathrm d\mathbf S=\mathbf n\,\mathrm dS=\dfrac{\mathbf s_v\times\mathbf s_u}{\|\mathbf s_v\times\mathbf s_u\|}\|\mathbf s_v\times\mathbf s_u\|\,\mathrm du\,\mathrm dv$
$\mathrm d\mathbf S=\mathbf s_v\times\mathbf s_u\,\mathrm du\,\mathrm dv$
$\mathrm d\mathbf S=36\langle\cos u\sin^2v,\sin u\sin^2v,\sin v\cos v\rangle\,\mathrm du\,\mathrm dv$

So we have the surface integral - which the divergence theorem says the above triple integral is equal to -

$\displaystyle\iint_{\partial E}\mathbf f\cdot\mathrm d\mathbf S=36\int_{v=0}^{v=\pi}\int_{u=0}^{u=2\pi}\mathbf f(x(u,v),y(u,v),z(u,v))\cdot(\mathbf s_v\times\mathbf s_u)\,\mathrm du\,\mathrm dv$
$=\displaystyle36\int_{v=0}^{v=\pi}\int_{u=0}^{u=2\pi}(12\cos u\cos v\sin^2v+6\sin^2u\sin^3v)\,\mathrm du\,\mathrm dv=288\pi$

as required.

3 0

3 years ago

A motorboat travels 9 miles downstream (with the current) in 30 minutes. The return trip upstream (against the current) takes 90

Ksju [112]

Answer:

Option B

Step-by-step explanation:

When moving with the current, the speeds of the boat and current will add up. The first situation can then be represented by:

9 = 0.5(x + y)

While opposing the current, the speeds will be subtracted. Forming:

9 = 1.5(x - y)

The second option contains this equation system, so it is correct.

8 0

2 years ago

The radius of a circle is doubled. Which of the following describes the effect of this change on the area? PLZZZ HELP!!! THE RED

goblinko [34]

Answer:

The area is multiplied by 4

Step-by-step explanation:

The area of a circle is directly proportional to its square radius so if the radius is increased by x, the area will be increased by x ^ 2

4 0

2 years ago

Read 2 more answers

Factor 54^3 -128 ( please factor

kolezko [41]

54x³ -128 = 2(27x³ - 64) = 2[ (3x)³ - (4)³ ] = 2(3x - 4)( 9x² + 12x + 16)

Answer: 2(3x - 4)( 9x² + 12x + 16)

5 0

2 years ago

Read 2 more answers