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Free_Kalibri [48]
3 years ago
10

Find the slope of the line. Write your answer in simplest form.

Mathematics
1 answer:
Savatey [412]3 years ago
3 0

The slope of this line is: m = 3/4

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Set up, but do not evaluate, the integral that represents the length of the curve given by x = 1 + 3t^2, y = 4 + 2t^3 over the i
kherson [118]

L

=

∫

t

f

t

i

√

(

d

x

d

t

)

2

+

(

d

y

d

t

)

2

d

t

. Since  

x

and  

y

are perpendicular, it's not difficult to see why this computes the arclength.

It isn't very different from the arclength of a regular function:  

L

=

∫

b

a

√

1

+

(

d

y

d

x

)

2

d

x

. If you need the derivation of the parametric formula, please ask it as a separate question.

We find the 2 derivatives:

d

x

d

t

=

3

−

3

t

2

d

y

d

t

=

6

t

And we substitute these into the integral:

L

=

∫

√

3

0

√

(

3

−

3

t

2

)

2

+

(

6

t

)

2

d

t

And solve:

=

∫

√

3

0

√

9

−

18

t

2

+

9

t

4

+

36

t

2

d

t

=

∫

√

3

0

√

9

+

18

t

2

+

9

t

4

d

t

=

∫

√

3

0

√

(

3

+

3

t

2

)

2

d

t

=

∫

√

3

0

(

3

+

3

t

2

)

d

t

=

3

t

+

t

3

∣

∣

√

3

0

=

3

√

3

+

3

√

3

=6The arclength of a parametric curve can be found using the formula:  

L

=

∫

t

f

t

i

√

(

d

x

d

t

)

2

+

(

d

y

d

t

)

2

d

t

. Since  

x

and  

y

are perpendicular, it's not difficult to see why this computes the arclength.

It isn't very different from the arclength of a regular function:  

L

=

∫

b

a

√

1

+

(

d

y

d

x

)

2

d

x

. If you need the derivation of the parametric formula, please ask it as a separate question.

We find the 2 derivatives:

d

x

d

t

=

3

−

3

t

2

d

y

d

t

=

6

t

And we substitute these into the integral:

L

=

∫

√

3

0

√

(

3

−

3

t

2

)

2

+

(

6

t

)

2

d

t

And solve:

=

∫

√

3

0

√

9

−

18

t

2

+

9

t

4

+

36

t

2

d

t

=

∫

√

3

0

√

9

+

18

t

2

+

9

t

4

d

t

=

∫

√

3

0

√

(

3

+

3

t

2

)

2

d

t

=

∫

√

3

0

(

3

+

3

t

2

)

d

t

=

3

t

+

t

3

∣

∣

√

3

0

=

3

√

3

+

3

√

3

=

6

√

3

Be aware that arclength usually has a difficult function to integrate. Most integrable functions look like the above where a binomial is squared and adding the two terms will flip the sign of the binomial.    

Be aware that arclength usually has a difficult function to integrate. Most integrable functions look like the above where a binomial is squared and adding the two terms will flip the sign of the binomial.

8 0
3 years ago
What is 0.5854 as a percent
Vikki [24]
Move the decimal over by 2
so 58.54%
that simple
8 0
3 years ago
Read 2 more answers
I really need help who can help me?
Gelneren [198K]
Yes
Y varies directly with x, it is a direct relationship between the two.
3 0
3 years ago
Determine the greatest common factor of 15xy^7z and 30x^2y^7z
yawa3891 [41]

Answer:

15x {y}^{7} z

Step-by-step explanation:

<h3>Factorize the Expressions</h3>

15x {y}^{2} z = 3 \times 5 \times x \times y \times y \times y \times y \times y \times y \times y \times z

30 {x}^{2}  {y}^{7} z = 2 \times 3 \times 5 \times x \times x \times y \times y \times y \times y \times y \times y \times y \times z

<h3>Factor out the Common Factors</h3>

gcf = 3 \times 5 \times x \times y \times y \times y \times y \times y \times y \times y \times z

<h3>Multiply the Common Factors</h3>

3 \times 5 \times x \times y \times y \times y \times y \times y \times y \times y \times z = 15x {y}^{7} z

The Factored expressions are

15x {y}^{7} z

And

15x {y}^{7} (2x)

5 0
3 years ago
to find the area of a trapezoid dylan uses the formula A=1/2 (B1+B2)h the bases have lengths of 3.6 cm and 12 1/3 cm. the height
Artemon [7]

Answer:

It is irrational because it can not be represented as a fraction of two integers

Step-by-step explanation:

Given

B_1 = 3.6

B_2 = 12\frac{1}{3}

H = \sqrt[3]5

Required

Why is the area irrational?

First, we need to calculate the area

Area = \frac{1}{2}(B_1 + B_2) * H

Area = \frac{1}{2}(3.6 + 12\frac{1}{3}) * \sqrt[3]5

Area = \frac{1}{2}(3.6 + \frac{37}{3}) * 17100

Area = \frac{1}{2}(\frac{10.8+37}{3}) * 1.710

Area = \frac{1}{2}(\frac{47.8}{3}) * 1.7100

Area = \frac{47.8}{6} * 1.7100

Area = \frac{47.8* 1.7100}{6}

Area = \frac{81.738}{6}

Area = 13.623

<em>It is irrational because it can not be represented as a fraction of two integers</em>

5 0
3 years ago
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