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4 years ago

What is the number 2,305,012 written in expanded notation?

Mathematics

2 answers:

Semenov [28]4 years ago

8 0

The expanded form of that number is 2,000,000 + 300,000 + 5,000 + 10 + 2.

OlgaM077 [116]4 years ago

3 0

2,000,000 + 300,000 + 5,000 + 10 + 2 = 2,305,012

I hope that this helped you :3

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

const2013 [10]

Answer:

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

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

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

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

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

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

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

a group of friends went to lunch. the bill, before sales tax and tip, was $37.50. a sales tax of 8% was added. the group then ti

valentina_108 [34]

First, there was $37.50.
Then, there was a tax of 8%.
Then, there was a tip of 18%.
Since the tip is after the tax, it's not going to affect it, and we can essentially ignore it.

The real question here is just what is 8% of $37.50?

To find the percent of something, first you need to put the percent as a decimal.

8% in decimal form is 0.08 (If you think about it literally, they're both 8 hundredths)

Finally, just multiply the percent in decimal form by the number.

37.50 dollars × 0.08 = 3 dollars.

6 0

3 years ago

Read 2 more answers

It costs Sally 20 cents to make a cup of lemonade, and she sells each cup for 50 cents. She is using the following identity to c

mote1985 [20]

She would make 3.00 dollars

50(10) - 20(10) = (10)(50 - 20)

500 - 200 = (10 * 30)

300 = 300

(10)(50 - 20) = 30(10)

(10 * 30) = 300

300 = 300