Questions on Geometry: Circles and their properties answered by real tutors!

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What is the base length of an isosceles triangle that has two legs of 5 inches and the obtuse angle of 120 degres and the two base angles of 30 degress?
----------------------
If you sketch this, then draw the altitude from the upper vertex (=120 degs), you see that 5 is the hypotenuse of each of 2 congruent triangles. Half of the base is 5*cos(30) = 5*sqrt(3)/2. So the base is 5*sqrt(3), = ~8.6603

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A building is constructed over a circular pond.
The ratio from the longest side of the rectangular building is 4:3 and the perimeter is 280 units.
---
4x+3x = 280
7x = 280
x = 40
length = 3x = 120 units
width = 4x = 160 units
-----
diagonal = sqrt[120^2+160^2] = 200 units
--------
Area = 120*160 = 19200 sq. units
------------------------------------
The edge of the pond is 4 units beyond each corner of the building on the diagonal when viewed in an aerial photo.
diameter of the pond = 200 +2*4 = 208 units
area of the pond = 104^2(pi) = 10816pi sq. units
----------------------------
The building owners plan to put water plants in the area of the pond that will recieve sunlight at noon.
That area = pond area - rectangle area = 10816pi - 19200 = 14779.47 sq. units
-----------------
The plants cost $1.25 each and require a surface area of 30 square units to be healthy when mature. What is the total cost of the plants needed to cover the available area?
cost = [14779.47/30]*1.25 = 492.65*1.25 = $615.81
========================
Cheers,
Stan H.

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A moving circle is tangent to the x-axis and to the circle of radius one with center at (2,-6). Find the equation of the locus of the center of the moving circle.
----------------
I have a solution, but I'll have to email some drawings and hand-written notes.
Send you email address to gsihoutx@aol.com and I'll send it tomorrow.
--------------------

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find the perimeter and the area the length of this 1/4 is 3 cm
:
Assume the radius is 3 cm
:
If you look a 1/4 circle you can see the perimeter will be:
1/4 * circumference + 2* radius
:
P = + 2r
:
P = + 2(3)
:
P = 4.71 + 6
:
P = 10.71
:
:
Area: find the area of the circle and divide it by 4
:
A =
:
A =
:
A = 7.068 on my calc

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AC is equal to the radius 20.
We know that angle C is half of the original angle of the original equilateral triangle(60) ....so C is 60/2=30. ABC is a right triangle so we know that angle A is a 60 degree angle 180-90-30=60
:
BC is equal to 1/2 of the entire side of the equilateral triangle
and we know that sine 60 degrees =BC/20(hypothenuse of the ABC)--->solving for BC=20(sine60degrees)=17.32
:
now if we double BC we will have the entire length of one side of the equilateral triangle ( 17.32(2)=34.64)---> and since all sides are equal the Perimeter is three times this length. 34.64(3)=103.92 inches

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GIVEN: linePS and lineQT are chords intersecting at R.
PROVE: triangleQRP is similiar to triangle SRT.
STATEMENTS
Comment: Draw the picture so you can see why the reasons are what they are.
-------------------------------------------------------------------------------
1. line PS and lineQT are chords intersecting at R.
1. Reason: Given.
-------------------------------
2. angle QPS is congruent to angle STQ and
angle PQT is congruent to angleTSP.
2. Reason: Each angle pair subtends the same arc and therefore
mQPS=mSTQ and mPQT=mTSP.
3. angle QRP = angle SRT.
3. Reason : these are vertical angles
4. triangle QRP is similiar to triangle SRT.
4: Reason: three angles of one triangle are correspondingly equal to three angles of the other triangle.
I just need the reasons for numbers 2 and 3.
===========================================
Cheers,
Stan H.

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If the area of a circle is 25 meters, what is its radius in meters?
.
Area of a circle is (pi)r^2
where
pi is a constant 3.14
r is radius
.
Plugging in the given values:
25 = (3.14)r^2
25/3.14 = r^2
25/3.14 = r^2
7.9618 = r^2
2.8217 meters = r (radius)

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Will someone PLEASE HELP!!!
write an equation for the circle with center (-4, sq. root 7) and radius of 5 units.
------------------
The general eqn of a circle is:
where the center is at (a,b) and the radius is r.
For the one you entered, it's

That can be manipulated and shown as:

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The length of arc AB is 6(pi)cm and the measurement of arc AB is 120 degrees. What is the diameter of the circle?
--------------------
The circle has 3(120) degrees
--------------------------------
The circumference of the circle is 3[6(pi)cm] = 18(pi)cm
----------------------
But circumference = (pi)d, where d is the diameter of the circle.
------------------------
EQUATION:
(pi)d = 18(pi)
d = 18 cm
==============
Cheers,
Stan H.

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Given: (3,-1) and (-11,7) are endpoints of a diameter of a cirle.
Find the radius, center coordinates, and equation in general form
------------------------------
Center Coordinates:
x coordinate: (3-11)/2 = -4
y coordinate: (-1+7)/2 = 3
Center: (-4,3)
--------------------------
Radius:
r = sqrt[(3--4)^2 + (-1-3)^2] = sqrt[49 + 16] = sqrt(65)
----------------------------
EQUATION:
(x+4)^2 + (y-3)^2 = 65
===========================
Cheers,
Stan H.

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P = r*angle (angle in radians)
------------------
120 degs = 2PI/3 radians
P = 7*2PI/3
P = apx 14.66 cm

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Edwin's solution:
For what nonzero value of the radius is the circumference of a circle numerically equal to its area?

If we want , then



Divide both sides by :



Cancel the 's





Get 0 on the right by adding  to both sides:



Factor out  on the left:



Use the zero-factor property:



Answer: the nonzero value of  is .

Edwin

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For what nonzero value of the radius is the circumference of a circle numerically equal to its area?
---------------------
Good question, I never thought of that.
Area = PI*r^2 = 2PIr = circumference
PI*r^2 = 2PIr
Divide by PI*r
r = 2
--------------
Area = 4PI
Circum = 4PI

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Let's graph the line:


 
We can draw two possible circles that touch that line and also the two
axes, a great big one and a tiny one: 



So we expect two solutions, both in quadrant 3.  

Let the radius of the circle be r, In either case the circle's center
will then be (-r,-r). 

If any circle with radius r in quadrant 3 touches both axes then 
the points (-r,0) and (0,-r) are the points where the
circle touches the axes, and the circle has center (-r,-r), and 
thus the equation of the circle is



This can be seen by drawing in these two radii,
since they are also the coordinates of the center (-r,-r)
(I'll just do it with the big circle.):



Now if we draw a third radius touching the line:



We know that the distance from a point 
 to a
line  is found by this equation:



So the distance from the line 

to the center (-r,-r) must also be equal to the radius r,
of the circle

So substituting 







Squaring both sides








Dividing through by 24


   

So the big circle has radius 2 and its equation becomes





And the little circle has radius 3/14 and its equation becomes





--------------------------

2." what is the equation of the circle who passes 
through the points (1,4),(7,5),(1,8)?"

Use the general equation of a circle:



Substitute the point (x,y) = (1,4)






Substitute the point (x,y) = (7,5)






Substitute the point (x,y) = (1,8)






Now we have a system of three equations in three
unknowns:



This has solution:



So the equation of the circle:

 becomes



Multiplying through by 2:



-----------------------------------

3."what is the equation of the circle who has its centre 
on the line and passes through the origin 
and the point (4,2)?"

We'll draw the line and mark the point (4,2):

  

Now since the circle must go through (0,0) and (4,2),
the line segment joining these two points must be a 
chord of the circle.  So we'll draw it.

 

Now the perpendicular bisector of a chord must pass through
the center of the circle, so let's get the equation of the
perpendicular bisector of that chord from (0,0) to (4,2).

First we need its slope.  So we find the slope of the chord,
take its reciprocal with the opposite sign.

Slope of the chord:






So the slope of its perpendicular bisector is  or .

The perpendicular bisector must also go through the midpoint of
the chord, so we use the midpoint formula:







So we find the equation of the perpendicular bisector of the
chord.  It has slope = m =  and it goes through (2,1).
So we use the point-slope formula:







So we'll draw this perpendicular bisector of the chord:

 

Where they intersect must be the center of the required
circle, so we solve the system of their equations:



Solving that system we get (x,y) = (4,-3).

So the center of the circle is (4,-3).

 

and we can sketch in the circle:



We have its center (4,-3).
Now we must find its radius, which is the
distance from the center (4,-3) to the point (4,2).

 


Using the distance formula:










So the radius is 5 and the center is (4,-3),
so the equation is





Edwin

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Here are the steps:
1-complete the square in both x and y to put the equation in standard form.
2-Group the expressions involving x
3-Group the expressions involving y
4-Put the constant on the right side of the equation
Follow the steps on your own.
If you cannot solve it, write back.

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Circumference of any circle is:
C = (pi)/d
where
C is circumference
d is diameter
.
Area of any circle:
A = (pi)r^2
where
A is area
r is radius
.
Of course, radius is half the diameter or
r = d/2
.
Pulling it all together:
Since they gave you circumference, use the circumference formula to find d (diameter):
C = (pi)/d
18(pi) = (pi)/d
18=d
.
Extract radius (r):
r = d/2
r = 18/2
r = 9 cm
.
plug it into the formula for area of a circle:
A = (pi)r^2
A = (pi)9^2
A = 81(pi) square cm (this is what they're looking for)

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AB, BE, and AE form a right triangle with the right angle at the point of tangency (B)

BE is the radius (r) of the Earth __ AE is the radius (r) plus the altitude of the satellite

using Pythagoras __ AB^2+BE^2=AE^2 __ 15977^2+r^2=(r+12500)^2 __ 15977^2+r^2=r^2+25000r+12500^2

subtracting r^2+12500^2 __ 15977^2-12500^2=25000r __ dividing by 25000 __ 3961=r (approx)

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In a circle whose diameter is 20 inches, a chord is 6 inches from the center. What is the length of the chord?
:
Here is one way. Draw a circle, draw in the chord. See that a triangle is formed
by length of the chord and two radii.
:
This triangle consists of two right triangles where:
hypotenuse = 10" (radius)
one side = 6" distance from chord to the center
one side = 1/2 the length of the chord (x)
:
x^2 + 6^2 = 10^2
x^2 + 36 = 100
x^2 = 100 - 36
x =
x = 8" is half the chord, therefore the chord = 16"

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Do I divide the circumference of a circle by four to get the perimeter of a quadrant?
-------------
Perimeter means distance around. You would only be
finding the length of the arc of the quadrant. I
think you would have to add 2 times the radius to get
the distance around the quadrant.
---------------
Cheers,
Stan H.

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5/2pi=pid
d=(5pi/2)/pi
d=5/2 answer.

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Like any other similar terms you add them:
45 degrees + 63 degrees=108 degrees.

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How many cm. does 10 inches equal?
----------------
An inch is 2.54 cm exactly. Not 2.539976... or whatever, it's 2.54 cm EXACTLY.
So if one inch is 2.54 cm, 10 inches is ten times as many, or
2.54 x 10
= 25.4 cm

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Here is the simple conversion from inches to centimeters:
1 inch=2.54cm

Therefore,
1(x) inches=2.54(x) cm

Plug in your numbers
1(10) inches= 2.54(10) cm

Simplify
10 inches=25.4 cm

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.
Note: E+11 is exponential notation for 10^11 (that's eleven zeros)
.
Let's take it one step at a time:
how would i find the surface area to compare?
SA=4(3.14159)r^2
The above requires r (radius) but the information in the problem provided diameters. But, we know that:
radius = d/2
88,846 miles sphere: radius = 88846/2 = 44423 miles
7,926 miles sphere: radius = 7926/2 = 3963 miles
.
SA of 88,846 miles sphere:
4(3.14159)(44423)^2
= 12.56636(44423)^2
= 12.56636(1973402929)
= 24798491630.86844 square miles
.
SA of 7,926 miles sphere:
4(3.14159)(3963)^2
= 12.56636(3963)^2
= 12.56636(15705369)
= 197359320.78684 square miles
.
Ratio of "SA of 88,846 miles sphere" to "SA of 7,926 miles sphere" then is
24798491630.86844:197359320.78684
= 24798491630.86844/197359320.78684
= 125.651 (this is the ratio)
.
Volume of 88,846 miles sphere:
(4/3)(pi)(44423)^3
= (4.18879)(44423)^3
= (4.18879)(87664478314967)
= 367208090120950.61993 cubic miles
.
Volume of 7,926 miles sphere:
(4/3)(pi)(3963)^3
= (4.18879)(3963)^3
= (4.18879)(62240377347)
= 260711870227.34013 cubic miles
.
Ratio of volumes:
367208090120950.61993:260711870227.34013
367208090120950.61993/260711870227.34013
1408.482

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V=455.9
What is the radius and diameter of the basketball?
-----------
Assuming the ball is a sphere, the volume of a sphere is (4/3)PIr^3
455.9 = 4*PI*r^3/3
r^3 = 455.9*3/(4*PI)
r^3 = 108.838
r = 4.77 units
d = 9.54 units

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Volume of a sphere = (4/3)(pi)r^3
.
The problem told us that the volume = 455.9
.
455.9 = (4/3)(3.14)r^3
1367.7 = (4)(3.14)r^3
341.925 = (3.14)r^3
341.925 = (3.14)r^3
108.893 = r^3
4.78 inches = r (radius)
.
Diameter = 2r = 2(4.78) = 9.55 inches (diameter)

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Can someone please list some good Christian songs, so I can put them on my ipod?
-------------
No, because there aren't any.

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As you know,
radius = diameter/2
AND, since the problem stated that the diameter is 1 ft:
radius = 1/2 ft
.
Plug it into:
V = 4/3 π r^3
V = 4/3 π (1/2)^3
V = 4/3 π (1/8)
V = 1/3 π (1/2)
V = 1/6 π
V = π/6

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Find an equation of the circle having center (4,-3) and radius 5?
---------------
Plot the point (4,-3) on an x/y coordinate system.
Draw a circle around it with radius approximately 5.
Pick any point on that circle and label it (x,y)
From that point draw a segment perpendicular to the line y=-3.
Draw a radius sement from (4,-3) to (x,y)
------------------
You should see a right triangle with vertices at
(4,-3), (x,y) and (x,-3)
That right triangle has hypotenuse = 5,
a side of length y+3 and a side of length x-4.
--------------------------
Using Pythagoras you have:
(x-4)^2 + (y--3)^2 = 5^2
or
(x-4)^2 + (y+3) ^2 = 5^2
That is the equation for every point on th circle.
==================
Cheers,
Stan H.

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Formula to use:
The perimeter of a circle with diameter D is pi*D
When the wheel turns one time, the vehicle will move a distance of pi*D, If it turns 10 times, the vehicle will cover a distance of 10*pi*D.
Therefore, The distance the vehicle traveled is:
10*pi*D
=10*3.14*28
= 879.2 inches
Converting it to foot,we have(Note that 1 feet = 12 inches, so 1 inch = 1/12 feet)
879.2 inches = 879.2*1 inches = 879.2*(1/12)foot=879.2/12 foot = 73.3 foot

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if the side of the square is 20mm, then the diameter of each circle is 20mm. That makes the radius of each circle 10mm. Now find the area of the circles (which would overlap in a square, so I guess I should say the circle).

area of circle=
area of circle=
area of circle=

If the circle is shaded, then that is the area of the shaded part of the square.

Area of shaded region=

If everything but the circles are shaded, find the area of the square.

area of square=
area of square=
area of square=

Now subtract the circle from that.

Area of shaded region=

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the segments GA and BG are tangent to a circle at A and B, and AGB is a 48 degree angle. Given that GA = 12 cm, find the distance from G to the nearest point on the circle.

 

Label the center of the circle O. and 
draw radii to A and B. Let the radius be r



Next draw OG which bisects the 48° angle G into
two 24° angles. Let P be the point where OG intersects
the circle.  P is the nearest point on the circle to G, 
so GP is the distance we're looking for.

Plan: Calculate the radius OA and the hypotenuse OG using the 
upper right triangle using trig ratios. Then calculate OG. Then 
since OP is also a radius, we will subtract the radius OP from 
OG and get GP.
 


In the right triangle AOG, radius AO is the side opposite
angle AGO which is 24°.  GA is the side adjacent to angle AGO.

So we use 







Put 1 under the 



Cross-multiply:



Next we calculate OG:

---
OG is the hypotenuse, GA is the opposite side of 24°

So we use 







Put 1 under the 



Cross-multiply:



So now we can find GP by

subtraction, since OP = r = 12tan(24)

GP = OG - OP = 

Edwin

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Find the equation of the circle described each.
a)The circle is tangent to both coordinate axes and contains the point (6,3).

First plot the point (6,3)



I can see how there could be two different solutions, by drawing in
these:



The equation of a circle with center (h,k) and radius r is



Draw in radii to the axes:



We can see that since the circle has to be tangent to both axes, 
its center has to have the same x and y coordinates. and also that
the radius has to be equal to h as well.  So we can see that all
three values h, k, and r, must all be the same. So let them all be 
h, i.e., h = k = r, and we have



So since , we have


 


Now since it contains the point (6,3) we can substitute that in











That simplifies to



Factoring:



 or 
 or 

So we two values of , so

the two circles' equations are

 and 

 and 



b)The circle is circumscribed about the triangle whose vertices are (-1,-3),(-2,4),and (2,1). 

We plot the points:



We draw the triangle:



The equation of a circle with center (,) and radius  is:



We substitute the point (,) = (,) 













Get all squared terms on right:



We substitute the point (,) = (,) 













Get all squared terms on right:



We substitute the point (,) = (,) 













Get all squared terms on right:



So we have the three equations:




 

Since the right sides of all three equations are equal,
then so are the left sides:



Using the first two





Using the first and third:





So we solve these two equations by 
substitution of elimination:




and get (h,k) = (,)

To find r we go back to



















Therefore the equation 



becomes





So we plot the center (,)



Now put the point of the compass on the center 
and draw the circle:



-----------------------

Maybe I'll do the last one tomorrow.  I'm getting sleepy. Check back to
see if I've done it.

Edwin

c)The sides of a triangle are on the line 6x+7y+11=0,2x-9y+11=0,and 9x+2y-11=0.Find the equation of the cirlce inscribed in the triangle.
(looking forward for someone who will asnwer this..ty in advance =)

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C=2*PI*R
C=2*3.14*7
C=43.96 IS THE CIRCUMFERANCE.
43.96/2=21.98 PLANTS OR ROUNDED 22 PLANTS WILL DO.

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Start with the circumference of a circle formula


Plug in


Multiply 2, and (you can use 3.14 for pi) to get



So the circumference of a circle with a radius of 4.1 meters is 25.8 meters (rounded to the nearest tenth)

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What is the circumference of a circle with a radius of 4.1 m? Round to the nearest tenth.Use pi=3.14
-----------------
Circumference = 2(pi)r
C = 2(3.14)4.1 = 25.748
===================
Cheers,
Stan H.

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Ok... if you make Mr. Naeher's height into feet by dividing 74 inches by 12 you'll get... 6ft 2"
If you divide Ms. Snyder's height of 62 inches by 12 you get 5'2"
Since Mr. Naeher is taller he doesn't have to look up as high as Ms. Snyder does to see the bird... therefore his angle of elevation is smaller than Ms. Snyder.
In this problem, the distance between the buildings is irrelevant.
Just a little common sense!! ;) Good luck!

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Container:
r = 4.8/2 = 2.4 inches
h = 6.5
volume of cylinder = h(pi)r^2
volume of cylinder = 6.5(pi)(2.4)^2
.
cone:
r = 1/2 d = 1/2(1.25) = 0.625 inches
volume of sphere = 4/3(pi)r^3
volume of sphere = 4/3(pi)(0.625)^3
.
# of scoops = "volume of cylinder"/"volume of sphere"
[6.5(pi)(2.4)^2]/[4/3(pi)(0.625)^3]
[6.5(2.4)^2]/[4/3(0.625)^3]
[37.44]/[0.32552]
115.016
.
or, approximately
115 scoops

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Volume of the cube:

Start with the volume equation.


Plug in the given dimensions.


Multiply



--------------

Volume of the sphere:




Plug in (note: since the side length is 4, this means that that the diameter is 4)


Evaluate the right side.




So to find the percentage, simply divide the volume of the sphere by the volume of the cube






Simplify


So the ratio of the volume of the sphere to the volume of the cube is


Note: it turns out that the ratio of the volume of the sphere to the volume of the cube is no matter what the dimensions of the cube are.



Now approximate to get . So the percentage of the volume of the sphere to the volume of the cube is 52.4%

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for a circle of the form x^2+ax+y^2+by+c=d; the coordinates of the center are [(-a/2),(-b/2)]

circle P is centered at (3,-2) and circle Q is centered at (4,-5)

if by difference, you mean distance; then d=sqrt[(4-3)^2+(-5-(-2))^2] __ d=sqrt(10)

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Find the total grazing area of a goat G, The animal is tied to a corner of a 40'x40' barn, by an 80' rope. One of the sides of the barn is extended by a fence. assume that the area is grass everywhere except inside of the barn.
;
The side that is extended by the fence will make a difference
:
If the fence is opposite the corner, where the rope is tied, then is will not effect it.
:
The larger area is of circle with 80' radius
A =
A = 15079.6 sq/ft
and
2 each circles with a radius of 40'
A =
A = 2513.3 sq/ft
:
Total: 15079.6 + 2513.3 = 17,593 sq/ft

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pi*r^2=A (circle)
pi*2r=C (circle)
C/s=2pi radians/s [s is in radians]
A {circle}*(s/(2pi radians))=A {enclosed by the circular sector}
.
Ed

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You could try proportions on this problem:
Let's compare the ratio of the arc length, S, to the circumference of the circle, C, with the ratio of the area enclosed by the sector to the area of the entire circle . Remember that

Simplify and solve for

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True or False. If the radius of a circle is doubled, the area increases by a factor of 4.
To answer this, you need to know the area of a circle is:
(pi)r^2
if you were to double the radius, it would be:
(pi)(r+2)^2
= (pi)(r+2)(r+2)
= (pi)(r^2+4r+4)
This increased the area by MORE than a factor of 4.
Anwer: False
.
*******************************************************
.
True or False. Increasing the radius of a circle by 10 inches, increases the circumference of the circle by 20 inches.
To answer this, you need to know the circumference of a circle is:
2(pi)r
Now, if we increase the radius by 10:
2(pi)(r+10)
= 2(pi)(r+10)
= 2(pi)(r+10)
= 2(pi)r+2(pi)10
= 2(pi)r+(pi)20
This increased the circumference by (pi)20.
Answer: False
.
*******************************************************
.
Find the perimeter of a region bounded on three sides by a rectangle with a length of 1000 ft and a width of 600 ft and bounded above by a semicirle with a radius of 300 ft. Use 3.14 as the approximation for ( Do not include the lower half of the semicircle in the perimeter)
A)3542 ft
B)3592 ft
C)3502 ft
D)3522 ft
Circumference of the semi-circle is:
(.5)2(pi)r
= (.5)2(3.14)300
= (.5)2(3.14)300
= 2(3.14)150
= 2(471)
= 942 ft
perimeter of the 3-sided rectangle:
1000+1000+600 = 2600 ft
total perimeter:
942 ft + 2600 ft = 3542 ft
Answer: A
.
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Find the perimeter of an equilateral triangle (all sides have equal lengths) with side lengths 4 mm
There are three sides to a triangle, therefore the perimeter is the total of the lengths:
4+4+4 = 12 mm

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What is the diameter of the circle whose area is 36PI
Area = (pi)r^2
(pi)r^2 = 36(pi)
r^2 = 36
r = 6
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Diameter = 2r = 12 units
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Cheers,
Stan H.