Questions on Word Problems: Geometry answered by real tutors!

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Question 751440: A pizza parlor has two different sizes of pizza. The smaller one has a diameter of 10 inches and the larger one has a diameter of 16 inches. What is the ratio of the area of the smaller pizza to the area of the larger pizza?
Answer by Cromlix(314)

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Smaller pizza has diameter of 10 ins. Radius = 5 ins.
Larger pizza has diameter of 16 ins. Radius = 8 ins
Area of smaller pizza = pi*r^2 => 78.54 ins^2
Area of larger pizza = pi*r^2 => 201.06 ins^2
78.54 : 201.06
1 : 2.56
hope this helps.
:-)

Question 751370: The volume in cubic metres of water in an aquarium is given by the polynomial v(x)=x^3-16x^2+79x-120.If the dept in metres can be represented by x-3, what are the possible dimensions of the rectangular aquarium in terms of x if the aquarium holds 70 cubic metres (70 000 litres)
The answer is 7 by 5 by 2 i need the steps though!!
ANY HELP APPRECIATED!
Answer by josgarithmetic(1507)

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If x-3 is actually a root for v(x), then you could more easily factor the function. Try synthetic division by (x-3). See what this allows you to do. I have not tried all this yet but it might be all you need.
...

I'm not getting anything like what you say is the answer. That "x-3" seems to be no help. At holding 70 cubic meters, v=x%5E3-16x%5E2%2B79x-120=70

;

Trying synthetic division according to Factor Theorem %28x-10%29%28x%5E2-6x%2B19%29=0

so a rational root is 10. The quadratic factor has imaginary parts.
x=10

if this aquarium holds 70 m^3, and the depth is x-3=10-3=7

meters.

Question 751082: What is the area of a circle with the circumference of 43.98226
Answer by Cromlix(314)

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Circumference/ pi = diameter (pi = 3.14)
43.98226/pi =14 units
Halve 14 for the radius = 7 units
Area of a circle = pi r^2
= 3.14 * 7^2
= 153.86 units^2

Question 751203: The perimeter of the rectangle equals the sum of the lengths of its four sides. If the width of a rectangle is 6 inches and its perimeter is 40 inches, find the length of the rectangle.
Answer by Cromlix(314)

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Perimeter = 2 Lengths + 2 Widths
If the width of the rectangle is
6 inches, then the perimeter minus 2 widths
= 28 inches. Divide 28 by 2 and you have the length
of the rectangle = 14 inches.

Question 751193: The length of a rectangle is 5 inches more than the width. The perimeter is 70 inches. Find the length .
Answer by tommyt3rd(528)

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we can use the perimeter to deduce that l+w=35 (how?)
and since l is the longer, l-5=w

now we find l,,,
l+(l-5)=35
2l=40
l=20

:)

Question 751152: A contractor is building a circular swimming pool. The radius of the circle 12 feet. He needs to know the circumference in order to buy tile to put around the edge of the pool. What is the circumference if he uses pi = 3.14
Answer by Cromlix(314)

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Circumference = pi x diameter
If radius = 12 ft, the diameter = 24 ft.
Circumference = 3.14 x 24
Circumference = 75.36 feet
Hope this helps
:-)

Question 751033: √(a - 3)/(4)=2
Answer by lwsshak3(6494)

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√(a - 3)/(4)=2
√(a - 3)=8
a-3=64
a=64+3=67

Question 751035: the length of a room is three and half times its width. if the perimeter of the floor of the room is 90 m , find the length and width of the room.

Answer by dkppathak(29)

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let length is L
width is B
L= 7/2 B
P =2(L+B)
as per condition
90= 2(7/2B+B)
90=9/2 B
B = 90x2/9 =20m
length =7/2 of 20 =70m
ANSWER
L=70m
B=20m

Question 751009: The first problem on this page illustrates an important relationship between perimeter and area. Let’s change the problem slightly so that it applies to many cases at once. Instead of 60 feet, let’s say you have 4x feet of fencing, where x is any number. What, in terms of x, should the dimensions of the largest rectangular enclosure be, whose perimeter is 4x feet? What would the area be?
Answer by stanbon(57307)

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The first problem on this page illustrates an important relationship between perimeter and area. Let’s change the problem slightly so that it applies to many cases at once. Instead of 60 feet, let’s say you have 4x feet of fencing, where x is any number. What, in terms of x, should the dimensions of the largest rectangular enclosure be, whose perimeter is 4x feet? What would the area be?
--------
perimeter = 2(L+W) = 4x
L+W = 2x
L = (2x-W)
-------
Area = L*W
A = (2x-W)W
----
Area = -W^2 + 2xW
-----
Cheers,
Stan H.
------

Question 750920: The area of a patio is 90 square feet. The length of the patio is 1 foot longer than the width. What is the length of the patio?
Answer by jim_thompson5910(28593)

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x(x+1) = 90

x^2+x = 90

x^2+x-90 = 0

(x+10)(x-9) = 0

x+10=0 or x-9=0

x=-10 or x = 9

Ignore the negative solution.

The only solution that makes sense is x = 9, so x+1 = 10

The length is 10 ft.

Question 750867: A rectangular parking lot covers an area of 1150 square feet. It is 20 feet longer than it is wide. An equation appropriate for finding the dimensions of the lot is:
Answer by JoeTaxpayer(112)

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We can solve this with just one variable.
The short side is X, the long side is X+20
(X)(X+20)=1150
X%5E2%2B20X=1150

The quadratic solver yields

Solved by pluggable solver: Quadratic Formula

Let's use the quadratic formula to solve for x:

Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29

So lets solve x%5E2%2B20%2Ax-1150=0

( notice

, and

)

x+=+%28-20+%2B-+sqrt%28+%2820%29%5E2-4%2A1%2A-1150+%29%29%2F%282%2A1%29

Plug in a=1, b=20, and c=-1150

x+=+%28-20+%2B-+sqrt%28+400-4%2A1%2A-1150+%29%29%2F%282%2A1%29

Square 20 to get 400

x+=+%28-20+%2B-+sqrt%28+400%2B4600+%29%29%2F%282%2A1%29

Multiply

to get

x+=+%28-20+%2B-+sqrt%28+5000+%29%29%2F%282%2A1%29

Combine like terms in the radicand (everything under the square root)

x+=+%28-20+%2B-+50%2Asqrt%282%29%29%2F%282%2A1%29

Simplify the square root (note: If you need help with simplifying the square root, check out this solver)

x+=+%28-20+%2B-+50%2Asqrt%282%29%29%2F2

Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%28-20+%2B+50%2Asqrt%282%29%29%2F2

Now break up the fraction

x=-20%2F2%2B50%2Asqrt%282%29%2F2

Simplify

So the solutions are:

x=-10%2B25%2Asqrt%282%29

And the first answer is positive. The second side is 20 feet longer.

Question 750907: Find two consecutive integers that have a product of 182.
Answer by jim_thompson5910(28593)

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x(x+1) = 182

x^2 + x = 182

x^2 + x - 182 = 0

Use the quadratic formula to solve for x

x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-%281%29%2B-sqrt%28%281%29%5E2-4%281%29%28-182%29%29%29%2F%282%281%29%29

Plug in

x+=+%28-1%2B-sqrt%281-%28-728%29%29%29%2F%282%29

x+=+%28-1%2B-sqrt%281%2B728%29%29%2F%282%29

So the two numbers are either

13 and 14

or

-14 and -13

Question 750884: The sum of the squares of two consecutive positive numbers is 41. What is the smaller number?
Found 2 solutions by stanbon, jim_thompson5910:
Answer by stanbon(57307)   (Show Source):
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The sum of the squares of two consecutive positive numbers is 41. What is the smaller number?
-------
1st: x
2nd: x+1
-----
Equation:
x^2 + x^2+2x+1 = 41
----
2x^2 + 2x - 40 = 0
x^2 + x -20 = 0
(x+5)(x-4) = 0
-----
Positive solution:
x = 4
x+1 = 5
=============
Cheers,
Stan H.
=============

Answer by jim_thompson5910(28593)   (Show Source):
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x^2 + (x+1)^2 = 41

x^2 + x^2 + 2x+1 = 41

2x^2 + 2x+1 = 41

2x^2 + 2x+1 - 41 = 0

2x^2 + 2x - 40 = 0

2(x^2 + x - 20) = 0

2(x + 5)(x - 4) = 0

x + 5 = 0 or x - 4 = 0

x = -5 or x = 4

Ignore -5 (since the two numbers are both positive). So one number is 4 and the other number is 5.

Answer: The smaller number is 4.

Question 750883: The length of a rectangle is 8 inches greater than it' width. Find the width of a rectangle if it's area is 105 square feet.
Answer by jim_thompson5910(28593)   (Show Source):
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A = LW

105 = (x+8)(x)

105 = x^2 + 8x

x^2 + 8x = 105

x^2 + 8x - 105 = 0

(x + 15)(x - 7) = 0

x + 15 = 0 or x - 7 = 0

x = -15 or x = 7

Toss out x = -15

So the only solution that makes sense is x = 7

Width: 7

Length: 15

Question 750872: ha ha what is 5+4*8-9
Answer by jim_thompson5910(28593)   (Show Source):
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Use PEMDAS

5+4*8-9 ... Start with the given expression.

5+32-9 ... PEMDAS (M for Multiplication)

37-9 ... PEMDAS (A for Addition)

28 ... PEMDAS (S for Subtraction)

=======================================================

So 5+4*8-9 = 28

Question 750749: A square yard has a 6-foot wide sidewalk around a square lawn. the sides of the yard, including the sidewalk, have length y. evaluate for y=70, the polynomial that gives the area of the sidewalk.
I have driven myself in circles on this problem. i originally found the area of the sidewalk to be 420 by multiplying 6*70 but upon further evaluation concluded that was incorrect by finding the area of the yard (70*70) to be 4900 and the area of the lawn (64*64) to be 4096 making the sidewalk take up 804 sqft of space. I am to a point where im thoroughly confused.
Answer by rothauserc(206)   (Show Source):
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A square yard has a 6-foot wide sidewalk around a square lawn. the sides of the yard, including the sidewalk, have length y. evaluate for y=70, the polynomial that gives the area of the sidewalk.
the area of the square including the sidewalk is y^2
the area of the sidewalk is y^2 - (y-12)^2
y^2 - (y^2 -24y +144)
y^2 -y^2 +24y -144
we are left with
24y - 144 = 24*70 -144 = 1536

Question 750789: Find an equation of variation where y varies directly as x and y = 8 when x = 2, find y when x = 10. Can you explain how to do these types of problems?
Answer by Cromlix(314)   (Show Source):
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y varies directly as x
To get y to equal x you require
a constant of proportionality = k
So that y = kx
To find k use the formula:
y = kx
Input the y value and the x value:
8 = k 2
Therefore k = 8/2 or 4
Put k into your formula
y = 4 x
Now to find y input the
new 'x' value
y = 4 (10)
so y = 40.
Hope this helps
:-)

Question 750541: johanna built a rectangular fence inside her garden that enclosed 102 square feet of land.If the length and width were both whole numbers which shows the least number of feet of fence johanna could have used

Found 2 solutions by josmiceli, JoeTaxpayer:
Answer by josmiceli(9671)   (Show Source):
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You can try possibilities:

The perimeter =
ft
-----------------------

ft
----------------------

ft
---------------------
It looks like 46 ft is the least she can buy

Answer by JoeTaxpayer(112)   (Show Source):
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The greatest area a fence can surround in a rectangle is in h shape of a square. Let me offer a reason why.
If one side is a tiny bit longer, the adjacent side is shorter by the same amount.
(X-Y)(X+Y)=X^2-Y^2 so the maximum area is when Y= 0.
In your case, we can't have sqr(102) as a side, so with two sides adding to 51, I'll say 25x26 are the dimensions.

Question 750450: the cost of mailing a package varies directly as the weight. If a 20z package costs 44 cents to mail, how much does it cost to mail a 15 oz packages?
Answer by checkley79(3050)   (Show Source):
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15/20=x/.44
15x=20*.44
15x=8.80
x=8.80/15
x=.59 ans.

Question 750414: a 15 foot ladder leaning against a building touches the wall 12 feet above the ground. How far from the building is the bottom of the ladder
Found 2 solutions by Alan3354, tommyt3rd:
Answer by Alan3354(30993)   (Show Source):
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a 15 foot ladder leaning against a building touches the wall 12 feet above the ground. How far from the building is the bottom of the ladder
----------------
It's a 3-4-5 right triangle, times 3.
--> 9 feet.
==============

Answer by tommyt3rd(528)   (Show Source):
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c=15
b=12
a=?
this triangle is in the 3-4-5 family of right triangles so a=10 ft

:)

Question 750477: To funnel down a 6x6 square at a 70 degree angle
Answer by Alan3354(30993)   (Show Source):
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To funnel down a 6x6 square at a 70 degree angle
--------------
Do you have a question?

Question 750486: f(x) = x^2 find f(0), f(2), and f(-2)
Answer by jim_thompson5910(28593)   (Show Source):
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f(x) = x^2

f(0) = 0^2 ... replace all copies of x with 0

f(0) = 0

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

f(x) = x^2

f(2) = 2^2

f(2) = 4

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

f(x) = x^2

f(-2) = (-2)^2

f(-2) = 4

Question 750485: f(x) = x^2
Answer by jim_thompson5910(28593)   (Show Source):
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If you want to graph, then you would get this

Question 750432: What is the area of a regular octagon measuring 37 m on each side?
Answer by tommyt3rd(528)   (Show Source):
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The formula for the area of a regular n-gon is

where
s is the length of any side
n is the number of sides

which is about 6610.1 m^2
:)

Question 750413: The area of a rectangle is 250ft. You want the length to be 5ft longer than twice its width. What should the dimensions of the rectangle be.
Answer by tommyt3rd(528)   (Show Source):
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A=lw=250
and
l=2w+5

so we can write
(2w+5)w=250
or
2w^2+5w-250=0
which has w=10 as one solution
therefore l=2(10)+5=25
so 10ft x 25ft

:)

Question 750412: in a right triangle the length of the hypotenuse is 52 and the length of one leg is 48 whats the length of the other leg
Answer by Alan3354(30993)   (Show Source):
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in a right triangle the length of the hypotenuse is 52 and the length of one leg is 48 whats the length of the other leg
--------------

Question 750057: finding the perimeter of a 6sides hexagon with 12cm?

Answer by Alan3354(30993)   (Show Source):
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finding the perimeter of a 6sides hexagon with 12cm?
-----------
If you mean the sides are 12 cm, the perimeter = 6*12 cm

Question 750076: At 3 pm,a 10-foot pole casts a shadow of 8 feet on a horizontal ground.How long must be a rope be,in order to connect the top of the pole and the tip of the shadow a the time of observation?
Answer by Cromlix(314)   (Show Source):
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Pythagoras.
The pole and the shadow are two sides of
a right angled triangle. The rope makes
the hypotenuse.
Therefore:
(Length of pole)^2 + (length of shadow)^2
= (Length of rope )^2
(10)^2 + (8)^2 = (Length of rope)^2
(Length of rope)^2 = 164
Length of rope = square root of 164
Length of rope = 12.8feet
:-)

Question 750067: A tree casts a shadow 18 feet long. At the same time, a vertical rod 9 feet high casts a shadow 6 feet long. How tall is the tree?
Answer by Cromlix(314)   (Show Source):
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Similar triangles
if you consider the tree and the rod as
one ratio and the tree's shadow and the
rod's shadow as the other ratio.
Then you have:
x/9 = 18/6 (x being the tree)
The sides must match like for like
Cross multiply:
6x = 162
Divide both sides by 6
x = 27
Therefore the tree is 27 feet tall.
:-)

Question 750039: Find, to the nearest square yard, the area of a triangular plot of ground that measures 15 yards by 20 yards by 25 yards.
Answer by tommyt3rd(528)   (Show Source):
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We use Heron's formula

where

Now a=15, b=20, c=25

:)

Question 749579: What is the area of a square parcel of land if the so side length is an integer?
Answer by Alan3354(30993)   (Show Source):
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What is the area of a square parcel of land if the so [sic?] side length is an integer?
----------
Area = s^2 where s is the side lenght.
This is true if s is an integer, or not.

Question 749543: A 5'6" person walking down the street notices his shadow. If the angle of elevation from the tip of their shadow to the sun is 60°, what is the distance from the tip of the shadow to the top of his head (round to 2 decimal places)?
Answer by JoeTaxpayer(112)   (Show Source):
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A 60° triangle has the ratio of base = 1/2, height = (sqr3)/2 diagonal = 1
The shadow is the base in this case or 5'6"/(sqr(3)) = 5.5/1.732 = 3.175 ft
or (3'2.1")

Question 749474: Find the measures of two supplementary angles if 4 times the measure od one is 36 degrees more than twice the measure of the other.

Answer by stanbon(57307)   (Show Source):
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Find the measures of two supplementary angles if 4 times the measure od one is 36 degrees more than twice the measure of the other.
------
Equations:
x + y = 180
4x = 2y + 36
--------------------
Modify:
x + y = 180
2x - y = 18
-------
Add and solve for "x":
3x = 198
x = 66 degrees
y = 180-66 = 114 degrees
===============================
Cheers,
Stan H.
===============

Question 749285: The area of the clear panel in the top section is 15 feet square. What is the hieght of the top section?
Answer by Alan3354(30993)   (Show Source):
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The area of the clear panel in the top section is 15 feet square. What is the hieght of the top section?
=================
Not enough info

Question 749129: A rectangle has an area of 2 3/4 square yards and a width of 1/3 yards, what is the length of the rectangle

Answer by Cromlix(314)   (Show Source):
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Area = 2 3/4 yds^2 or 11/4 yds^2
If we divide the area by the width
the answer will be the length.
= 11/4 / 1/3
= 11 x 3/4 x 1
= 33/ 4
length = 8 1/4 yds.

Question 749188: A sandbox is 2 meters long and 3.5 meters wide. What is the perimeter of the sandbox in centimeters

Answer by Cromlix(314)   (Show Source):
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Perimeter = 2 x length + 2 x width
= 2 x 2 m + 2 x 3.5 m
= 4m + 7m
= 11m.= 1100cms.

Question 749141: Please help me solve this equation: A pizza parlor offers pizzas with diameters of 8 in, 10 in, and 12 in. Fine the aread of each size of pizza. Round to the nearest tenth.
Answer by Cromlix(314)   (Show Source):
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8 ins. pi*4^2 = 50.3 ins^2
10 ins. pi* 5^2 = 78.5 ins^2
12 ins. pi* 6^2 = 113.1 ins^2

Question 749182: A rectangular flower bed is 3 feet long and 8 feet wide. How many feet of fence will be needed to fence the flower bed
Answer by Cromlix(314)   (Show Source):
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The fence is termed the perimeter.
Perimeter = 2 x length + 2 x width
= 2 x 3 + 2 x 8
= 6 + 16
= 22feet

Question 749152: The revenue for a small company is given by the quadratic function r(t)=15t2+7t+830 where t is the number of years since 1998 and r(t) is in thousands of dollars. If this trend continues, find the year after 1998 in which the company's revenue will be $1412 thousand. Round to the nearest whole year.
Answer by josmiceli(9671)   (Show Source):
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Find when

( can't use the negative square root )

1998 + 6 = 2004

Question 749155: The area of a rectangular wall in the classroom is 216 square feet. Its length is 3 feet shorter than three times its width. Find the length and width of the wall of the classroom.
Answer by JoeTaxpayer(112)   (Show Source):
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L=3W-3
L*W=216
(3W-3)*W=216 (substituted the 3W-3 for L into second equation)
3W^2-3W=216 (distributed W through)
3W^2-3W-216=0 (subtract 216 from both sides)
W^2-W-72=0 (divide both sides by 3)

(W-9)(W+8)=0 (factored. Note, W cannot equal a negative number, we ignore -8)
W=9 L=(3*9)-3 = 24 (solved)

Question 749145: If the height of a rectangular prism is decreased by 1/3 then which statement is true?
A- Volume would decrease by 1/3
B- Volume would decrease by 1/6
C-Volume would decrease by 1/9
D- Volume would decrease by 1/27
Answer by timvanswearingen(103)   (Show Source):
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It's A.
The volume of a rectangular prism is V=l*w*h
The volume of the prism with decreased height is V=l*w*1/3*h
Rearranging, you have V=l*w*h*1/3
Since you are multiplying the original volume (l*w*h) by 1/3, the volume will be decreased by 1/3

Question 749103: If a room is 136 square feet in perimeter. If the length of the room is 20 feet, what is the width?
Answer by Cromlix(314)   (Show Source):
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Perimeter = 2 x Length + 2 x width
136 = 2 x 20 + 2 x width
136 = 40 + 2 x width
136 - 40 = 2 x width
96 = 2 x width
96/2 = width
48ft = width