Suppose that t = x+y/1-xy. Show that when x = 1/a and y = 2a-1/a+2, the value of t does not depend on the value of a.

Find the area of the helicoid (or spiral ramp) with vector equation r(u, v) = ucos(v) i + usin(v) j + v k, 0 ≤ u ≤ 1, 0 ≤ v ≤ 9π

Natasha2012 [34]

Let $H$ denote the helicoid parameterized by

$\mathbb r(u,v)=u\cos v\,\mathbf i+u\sin v\,\mathbf j+v\,\mathbf k$

for $0\le u\le1$ and $0\le v\le9\pi$ . The surface area is given by the surface integral,

$\displaystyle\iint_H\mathrm dS=\iint_H\|\mathbf r_u\times\mathbf r_v\|\,\mathrm du\,\mathrm dv$

We have

$\mathbf r_u=\dfrac{\partial\mathbf r(u,v)}{\partial u}=\cos v\,\mathbf i+\sin v\,\mathbf j$
$\mathbf r_v=\dfrac{\partial\mathbf r(u,v)}{\partial v}=-u\sin v\,\mathbf i+u\cos v\,\mathbf j+\mathbf k$
$\implies\mathbf r_u\times\mathbf r_v=\sin v\,\mathbf i-\cos v\,\mathbf j+u\,\mathbf k$
$\implies\|\mathbf r_u\times\mathbf r_v\|=\sqrt{1+u^2}$

So the area of $H$ is

$\displaystyle\iint_H\mathrm dS=\int_{v=0}^{v=9\pi}\int_{u=0}^{u=1}\sqrt{1+u^2}\,\mathrm du\,\mathrm dv$
$=\dfrac{9(\sqrt2+\sinh^{-1}(1))\pi}2$

5 0

3 years ago

How do you divide a fraction

My name is Ann [436]

There are 3 Simple Steps to Divide Fractions:
Step 1. Turn the second fraction (the one you want to divide by) upside down
(this is now a reciprocal).

Step 2. Multiply the first fraction by that reciprocal

Step 3. Simplify the fraction (if needed)

5 0

3 years ago

Maria and John have decided that once they live in a house, they want to have a pet. They go to an animal shelter and find sever

rosijanka [135]

Answer:

A. 0.47 ; B. 0.40 ; C. 0.73 ; D. 0.23 & 4.26

Step-by-step explanation:

Total Animals = Total Possible Outcomes = 8 + 12 + 6 + 5 + 11 = 42
Probability = No. of Possible Outcomes / No.of total outcomes
Odds in favour of event = Prob (Event) / Prob (No Event) ; Odds in against event = Prob (No Event) / Prob (Event)

A. Probability of choosing a Cat = All no. of cats / Total no. of animals

= ( 8 + 12 ) / 42 = 20 / 42 = 0.47

B. Probability of choosing a Common Cat or a Labrador Retriever :-

No. of common cats + no.of labrador retrievers / Total Animals

= (12 + 5) / 42 = 17 / 42 = 0.40

C. Probability of selecting non 'mixed breed' dog :-

No. of animals except mix breed dogs / Total Animals

= ( 8 + 12 + 6 + 5 ) / 42 = 31 / 42 = 0.73

D. Probability of choosing Siamese cat = Siamese Cat / Total Animals = 8 / 42 = 0.19 ; Probability Not choosing Siamese Cat = 1 - Prob (Siamese Cat) = 1- 0.19 = 0.81.

Odds in favour of choosing Siamese Cat :-

Prob (Siamese cat) / Prob (Not Siamese Cat) = 0.19 / 0.81 = 0.23

Odds against choosing Siamese Cat :-

Prob (Not Siamese Cat) / Siamese Cat = 0.81 / 0.19 = 4.26

4 0

3 years ago

Simplify the expression by combining like terms. 16+8−3+6−9

lyudmila [28]

Answer=18
First step combine the positive like terms
16+8+6
|
|
\/
30
Second step is to combine negative like terms with 30
30-3-9

When you do this you add 9 and 3 together since they are similar to each other so it equals 12 there for the equation becomes.....
30-12

Final answer 18

4 0

3 years ago

Solve 2/3=11/x round to the nearest tenth

Reptile [31]

2/3 = 11/x
(11)2/3 = 11/x(11)
11/3 = x
x = <span>3.66666666667

So your answer is ....
x = 3.7

Good luck! :))
</span>

8 0

3 years ago