Ask question

Ask question

All categories

2 years ago

15

Consider an n-sided regular polygon inscribed in a circle of radius r. Join the vertices of the polygon to the center of the cir

cle, forming n congruent triangles. Determine the central angle theta in terms of n. Show that the area of each triangle is (1/2)r^2sintheta. Find the limit of the area as n approaches infinity.

1 answer:

Fudgin [204]2 years ago

8 0

Answer:

a. Theta = 360/n ( angles around a point)

Step-by-step explanation:

n =Number of congruent triangles ( angles having the same angles and. Sides but not in the same direction).

This develops the assumption that each of the triangles created are equal and sides are equal.

Area of ∆ is given by 1/2 * b* h

Since all triangles are congruent b =h As they are all radius size

Area = 1/2 * (r *sin(theta)) * (r)

Area = 1/2*r^2sin(theta).

As n increases r *sin(theta) tends to 0

You might be interested in

Which of the following functions gives the radius, r(v), of a conical artifact that is 20 inches tall as a function of its volum

nalin [4]

By calculus or elementary geometry, we know that the formula for the volume of a cone, given pi (the constant), its height h and its radius r is:
$V=pi*h*r^2/3$
Lets solve this relation for the radius:
$3V=pi*h*r^2
$
$\frac{3V}{pi*h} =r^2$
$\sqrt{\frac{3V}{pi*h}} = r$
Now we have found a general formula that gives us the radius of a cone, given its volume and height. We need only substitute height (20in) and pi=3.14 to get the numerical equation: r=0.2185* $\sqrt{V}$

5 0

3 years ago

What is the sguare root of 54

riadik2000 [5.3K]

Sq root (54) =
sq root (9*5) =
3 * sq root (5)

8 0

2 years ago

Which is the solution to the inequality?

makvit [3.9K]

Answer:

b>3.133333333

Step-by-step explanation:

13/5<b-8/15

13/5+8/15<b

39+8/15<b

47/15<b

3.13333333<b

b>3.13333333

6 0

2 years ago

Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true

IgorLugansk [536]

Answer:

(a) 95% confidence interval for the true average porosity of a certain seam is [4.52 , 5.18].

(b) 98% confidence interval for the true average porosity of a another seam is [4.12 , 4.99].

Step-by-step explanation:

We are given that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation 0.75.

(a) Also, the average porosity for 20 specimens from the seam was 4.85.

Firstly, the pivotal quantity for 95% confidence interval for the population mean is given by;

P.Q. = $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$ ~ N(0,1)

where, $\bar X$ = sample average porosity = 4.85

$\sigma$ = population standard deviation = 0.75

n = sample of specimens = 20

$\mu$ = true average porosity

<em>Here for constructing 95% confidence interval we have used One-sample z test statistics as we know about population standard deviation.</em>

<u>So, 95% confidence interval for the true mean, </u> $\mu$ <u> is ;</u>

P(-1.96 < N(0,1) < 1.96) = 0.95 {As the critical value of z at 2.5% level

of significance are -1.96 & 1.96}

P(-1.96 < $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$ < 1.96) = 0.95

P( $-1.96 \times {\frac{\sigma}{\sqrt{n} } }$ < ${\bar X-\mu}$ < $1.96 \times {\frac{\sigma}{\sqrt{n} } }$ ) = 0.95

P( $\bar X-1.96 \times {\frac{\sigma}{\sqrt{n} } }$ < $\mu$ < $\bar X+1.96 \times {\frac{\sigma}{\sqrt{n} } }$ ) = 0.95

<u>95% confidence interval for</u> $\mu$ = [ $\bar X-1.96 \times {\frac{\sigma}{\sqrt{n} } }$ , $\bar X+1.96 \times {\frac{\sigma}{\sqrt{n} } }$ ]

= [ $4.85-1.96 \times {\frac{0.75}{\sqrt{20} } }$ , $4.85+1.96 \times {\frac{0.75}{\sqrt{20} } }$ ]

= [4.52 , 5.18]

Therefore, 95% confidence interval for the true average porosity of a certain seam is [4.52 , 5.18].

(b) Now, there is another seam based on 16 specimens with a sample average porosity of 4.56.

The pivotal quantity for 98% confidence interval for the population mean is given by;

P.Q. = $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$ ~ N(0,1)

where, $\bar X$ = sample average porosity = 4.56

$\sigma$ = population standard deviation = 0.75

n = sample of specimens = 16

$\mu$ = true average porosity

<em>Here for constructing 98% confidence interval we have used One-sample z test statistics as we know about population standard deviation.</em>

<u>So, 98% confidence interval for the true mean, </u> $\mu$ <u> is ;</u>

P(-2.3263 < N(0,1) < 2.3263) = 0.98 {As the critical value of z at 1% level

of significance are -2.3263 & 2.3263}

P(-2.3263 < $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$ < 2.3263) = 0.98

P( $-2.3263 \times {\frac{\sigma}{\sqrt{n} } }$ < ${\bar X-\mu}$ < $2.3263$ ) = 0.98

P( $\bar X-2.3263 \times {\frac{\sigma}{\sqrt{n} } }$ < $\mu$ < $\bar X+2.3263 \times {\frac{\sigma}{\sqrt{n} } }$ ) = 0.98

<u>98% confidence interval for</u> $\mu$ = [ $\bar X-2.3263 \times {\frac{\sigma}{\sqrt{n} } }$ , $\bar X+2.3263 \times {\frac{\sigma}{\sqrt{n} } }$ ]

= [ $4.56-2.3263 \times {\frac{0.75}{\sqrt{16} } }$ , $4.56+2.3263 \times {\frac{0.75}{\sqrt{16} } }$ ]

= [4.12 , 4.99]

Therefore, 98% confidence interval for the true average porosity of a another seam is [4.12 , 4.99].

7 0

2 years ago

How do you write 7258630 in word form

Aloiza [94]

7,258,630-

seven million, two hundred fifty-eight thousand, six hundred thirty :-)

8 0

2 years ago

Read 2 more answers