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2 years ago

Give at least five problems solving about area of a sector of a circle.

Mathematics

1 answer:

Katyanochek1 [597]2 years ago

5 0

See below for the examples of sectors and arcs of a circle

<h3>Area of sector of a circle</h3>

The area of a sector is calculated as:

$A = \frac{\theta}{360} * \pi r^2$ ---- when the angle is in degrees

$A = \frac{\theta}{2} *r^2$ ---- when the angle is in radians

Take for instance, we have the following problems involving sector areas

Calculate the area of a sector where the radius of the circle is 7, and

The central angle is 30 degrees
The central angle is π/12 rad
The central angle is 90 degrees
The central angle is π/4 rad
The central angle is 180 degrees

Using the above formulas, the sector areas are:

1. $A = \frac{30}{360}* \frac{22}{7} * 7^2 = 12.83$

2. $A = \frac{\pi}{12} * 7^2 = 12.83$

3. $A = \frac{90}{360}* \frac{22}{7} * 7^2 = 38.5$

4. $A = \frac{\pi}{2} * 7^2 = 38.5$

5. $A = \frac{180}{360}* \frac{22}{7} * 7^2 = 77$

<h3>Examples of arc length</h3>

The length of an arc is calculated as:

$L= \frac{\theta}{360} * 2\pi r$ ---- when the angle is in degrees

$L = r\theta$ ---- when the angle is in radians

Take for instance, we have the following problems involving arc lengths

Calculate the length of an arc where the radius of the circle is 7, and

The central angle is 30 degrees
The central angle is π/12 rad
The central angle is 90 degrees
The central angle is π/4 rad
The central angle is 180 degrees

Using the above formulas, the arc lengths are:

1. $L = \frac{30}{360}* 2 * \frac{22}{7} * 7 = 3.7$

2. $L = \frac{\pi}{12} * 7 = 3.7$

3. $L = \frac{90}{360}*2 * \frac{22}{7} * 7 = 11.0$

4. $L = \frac{\pi}{4} * 7 = 11$

5. $L = \frac{180}{360}*2 * \frac{22}{7} * 7 = 22$

<h3>Examples of the arcs of a circle</h3>

The examples include:

A parabolic path
Distance in a curve
Curved bridges
Pizza
Bows

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How to outline a proof

Sav [38]

I'll give you an example from topology that might help - even if you don't know topology, the distinction between the proof styles should be clear.

Proposition: Let
S
be a closed subset of a complete metric space (,)
(
E
,
d
)
. Then the metric space (,)
(
S
,
d
)
is complete.

Proof Outline: Cauchy sequences in (,)
(
S
,
d
)
converge in (,)
(
E
,
d
)
by completeness, and since (,)
(
S
,
d
)
is closed, convergent sequences of points in (,)
(
S
,
d
)
converge in (,)
(
S
,
d
)
, so any Cauchy sequence of points in (,)
(
S
,
d
)
must converge in (,)
(
S
,
d
)
.

Proof: Let ()
(
a
n
)
be a Cauchy sequence in (,)
(
S
,
d
)
. Then each ∈
a
n
∈
E
since ⊆
S
⊆
E
, so we may treat ()
(
a
n
)
as a sequence in (,)
(
E
,
d
)
. By completeness of (,)
(
E
,
d
)
, →
a
n
→
a
for some point ∈
a
∈
E
. Since
S
is closed,
S
contains all of its limit points, implying that any convergent sequence of points of
S
must converge to a point of
S
. This shows that ∈
a
∈
S
, and so we see that →∈
a
n
→
a
∈
S
. As ()
(
a
n
)
was arbitrary, we see that Cauchy sequences in (,)
(
S
,
d
)
converge in (,)
(
S
,
d
)
, which is what we wanted to show.

The main difference here is the level of detail in the proofs. In the outline, we left out most of the details that are intuitively clear, providing the main idea so that a reader could fill in the details for themselves. In the actual proof, we go through the trouble of providing the more subtle details to make the argument more rigorous - ideally, a reader of a more complete proof should not be left wondering about any gaps in logic.

(There is another type of proof called a formal proof, in which everything is derived from first principles using mathematical logic. This type of proof is entirely rigorous but almost always very lengthy, so we typically sacrifice some rigor in favor of clarity.)

As you learn more about a topic, your proofs typically begin to approach proof outlines, since things that may not have seemed obvious before become intuitive and clear. When you are first learning it is best to go through the detailed proof to make sure that you understand everything as well as you think you do, and only once you have mastered a subject do you allow yourself to omit obvious details that should be clear to someone who understands the subject on the same level as you.

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3 years ago

The sum of two numbers is 17. The difference of the numbers is 3

maxonik [38]

10 and 7 are the correct answers!

7 0

3 years ago

John has decided to get in shape for the new year. He is planning to joining the local fitness club. The fitness club charges cu

Marianna [84]

110+(14.95 x 12)= $289.40

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3 years ago

If a hurricane was headed your way, would you evacuate? The headline of a press release issued January 21, 2009 by the survey re

Nikolay [14]

Answer:

The 98% confidence interval for population proportion of people who refuse evacuation is {0.30, 0.33].

Step-by-step explanation:

The sample drawn is of size, <em>n</em> = 5046.

As the sample size is large, i.e. <em>n</em> > 30, according to the Central limit theorem the sampling distribution of sample proportion will be normally distributed with mean $\hat p$ and standard deviation $\sqrt{\frac{\hat p (1-\hat p)}{n} }$ .

The mean is: $\hat p=0.31$

The confidence level (CL) = 98%

The confidence interval for single proportion is:

$CI_{p}=[\hat p-z_{(\alpha /2)}\times\sqrt{\frac{\hat p (1-\hat p)}{n} },\ \hat p+z_{(\alpha /2)}\times\sqrt{\frac{\hat p (1-\hat p)}{n} }]$

Here $z_{(\alpha /2)}$ = critical value and <em>α </em>= significance level.

Compute the value of <em>α</em> as follows:

$\alpha =1-CL\\=1-0.98\\=0.02$

For <em>α</em> = 0.02 the critical value can be computed from the <em>z</em> table.

Then the value of $z_{(\alpha /2)}$ is ± 2.33.

The 98% confidence interval for population proportion is:

$CI_{p}=[\hat p-z_{(\alpha /2)}\times\sqrt{\frac{\hat p (1-\hat p)}{n} },\ \hat p+z_{(\alpha /2)}\times\sqrt{\frac{\hat p (1-\hat p)}{n} }]\\=[0.31-2.33\times \sqrt{\frac{0.31\times(1-0.33)}{5046} },\ 0.31+2.33\times \sqrt{\frac{0.31\times(1-0.33)}{5046} } ]\\=[0.31-0.0152,\ 0.31+0.0152]\\=[0.2948,0.3252]\\\approx[0.30,\ 0.33]$

Thus, the 98% confidence interval [0.30, 0.33] implies that there is a 0.98 probability that the population proportion of people who refuse evacuation is between 0.30 and 0.33.

7 0

3 years ago

What is the sum of 2equal numbers

meriva

Let the number be x
The sum of 2 equal numbers is simply x+x or 2x.

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4 years ago