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3 years ago

Factor the following completely. r^2 - 14r -50

Mathematics

1 answer:

algol133 years ago

7 0

You can't factor it at all

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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

The weight of a tree is doubling every 3 years and how many years will be way of the tree be 1600% of its weight now

BARSIC [14]

Doubling every 3 years, meaning 200% every 3 years.

$1600 \div 200 = 8$
$8 \times 3 = 24$
The answer is 24 years.

Hope this helps. - M

6 0

3 years ago

Suppose that the hourly wages of fast food workers are normally distributed with an unknown mean and standard deviation. The wag

Nutka1998 [239]

Answer: 1.303639

Step-by-step explanation:

The t-score for a level of confidence $(1-\alpha)$ is given by :_

$t_{(df,\alpha/2)}$ , where df is the degree of freedom and $\alpha$ is the significance level.

Given : Level of significance : $1-\alpha:0.80$

Then , significance level : $\alpha: 1-0.80=0.20$

Sample size : $n=40$

Then , the degree of freedom for t-distribution: $df=n-1=40-1=39$

Using the normal t-distribution table, we have

$t_{(df,\alpha/2)}=t_{39,0.10}=1.303639$

Thus, the t-score should be used to find the 80% confidence interval for the population mean =1.303639

8 0

3 years ago

Help please! !!!!! !!!!

zhannawk [14.2K]

Answer:

I would assume it's all but the second one (earned $60 over the summer=false)

4 0

3 years ago

Read 2 more answers

Pls i really need help asap plsss help plsss plsss plsss help me i really need help and don't just put random answers put real a

Schach [20]

Answer:

The simple solutions are:

Right 6: (-2+6,-3) = (4,3)

Left 6: (-2-6,-3) = (-8,-3)

Up 6: (-2,-3+6) = (-2, 3)

Down 6: (-2,-3-6) = (-2,-9)

Those are for horizontal or vertical

line segments.

Step-by-step explanation:

4 0

3 years ago

Read 2 more answers