Areas and Volumes



Question: The base of a triangle is 3 more than twice the  height. If the area of triangle is 10 square centimeters, find the dimension of the triangle?

 Solution: 

 Let x be the height of triangle.
then the base of triangle is 2x+3.
Given Area of triangle =10 sq cm
Therefore Area of triangle

A=1/2 base*height
10=1/2(2x+3)*x
20=(2x+3)*x
20=2x^2+3x
2x^2+3x-20=0
2x^2+8x-5x-20=0
2x(x+4)-5(x+4)=0
(2x-5)(x+4)=0
Ignore x+4=0
x=-4
Consider 2x-5=0
x=5/2  cm height of triangle
base= 2x+3=2*5/2+3=8 cm base of the triangle.

Verification: A=(1/2)*(5/2)*8=10

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Question: A rectangular box with height 3 cm has a volume  of 24 cubic cms. The length of box is 2 more than the width, find the dimensions and volume of the box?

 Solution:

Let x be the width of rectangular box.
then the length of rectangular box is x+2.
Given height =3 cm
and Volume =24 cubic cms
Therefore Volume of Rectangular box

V=Length*Width*height
24=x*(x+2)*3
8=x^2+2x
x^2+2x-8=0
x^2+4x-2x-8=0
x(x+4)-2(x+4)=0
(x-2)(x+4)=0
Ignore x+4=0
x=-4
Consider x-2=0
x=2  cm width of Rectangular box
length= x+2=2+2=4 cm length of the rectangular box.

Verification: V=2*4*3=24.



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Question: A rectangular garden  is 10 m by 20 m. the owner, who has enough pine bark  to cover 136 sq m, wants a pine bark path  around the outside  of the garden. Find the width  of the path and the outside dimensions.?

 Solution: 

Let x =width of the path
2x+10=total outside width
2x+20=total outside length
10*20=200 m^2= Inner area
136 m^2= Border Area (pine bark)

Equation:

Total Area= Inner area + Border Area
(2x+10)(2x+20)=200+136
4x^2+40x+20x+200=200+136
4x^2+60x=136
x^2+15x=34
x^2+15x-34=0
x^2+17x-2x-34=0
x(x+17)-2(x+17)=0
(x-2)(x+17)=0
Ignore (x+17)=0
x=-17
consider x-2=0
x=2 m width of the path
2x+10=2(2)+10=14 m total outside width
2x+20=2(2)+20=24 m Total outside length
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Question: The volume of cone is 36π cubic centimeters. the radius of the base is twice the height. Find the dimensions of the cone?

 Solution:
 Let x be the height of the cone
radius r= 2x
Given volume of cone is 36π

Therefore Volume of cone V= 1/3*π*r^2*h
36π=1/3*π*(2x)^2*x
9=1/3*x^3
x^3=27
x=27 cm height of the cone.
radius r=2*27=54 cm radius of the cone.

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 Question: The base of the triangle is 3 less than the twice of the height. If the area of the triangle is 10 sqcm, find the dimensions of the triangle/

 Solution: 
Let x be the height of the triangle
base of the triangle is 2x-3
Given area of the triangle is 10 sqcm
we know the Area of triangle

A= 1/2*base*height
10=1/2*(2x-3)*x
20=2x^2-3x
2x^2-3x-20=0
2x^2-8x+5x-20=0
2x(x-4)+5(x-4)=0
(2x+5)(x-4)=0
x=4 cm height of triangle
2x-3 =5 cm base of triangle.

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Question: The height of parallelogram is 3 less than the base. if the area of the parallelogram is 40 sq m, find the dimensions of the parallelogram?

 Solution:

Let x be the base of parallelogram
then x-3 is the height of the parallelogram.
given Area of parallelogram is 40 sqm

Area of parallelogram
A=base*height
40=x*(x-3)
40=x^2-3x
x^2-3x-40=0
x^2-8x+5x-40=0
x(x-8)+5(x-8)=0
(x+5)(x-8)=0
x-8=0
x=8 m base of parallelogram
x-3=8-3=5 m height of parallelogram.

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Question: A rectangular box with height 2 cm has a volume of 50 cubic cm. The length of the box is 5 less than twice the width . Find the dimensions of the box?

 Solution:

 Let x be the width of rectangular box
then 2x-5 is the length of the rectangular box
given Volume of the rectangular box=50 cubic cm
height = 2 cm
Volume of rectangular box
V= length*width* height
50=(2x-5)*x*2
25=2x^2-5x
2x^2-5x-25=0
2x^2-10x+5x-25=0
2x(x-5)+5(x-5)=0
(2x+5)(x-5)=0
x-5=0
x=5 cm width of the rectangular box
2x-5=10-5=5 cm length of the rectangular box


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Question: A rectangular box with height 4 meters  has a volume of 576 cubic meters and a square base. Find the dimensions of the box.

 Solution: 

Let x be the length of the box.
given that box has got a square base it means length=base
base=x
given volume as 576 cubic meters and height as 4 meters.

Therefore volume of rectangular box

V=length*base*height
576=x*x*4
x^2=144
x=12 m length of the box
and 12 m base of the box.

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Question: A rectangular garden  is 10 m by 20 m. the owner, who has enough pine bark  to cover 216 sq m, wants a pine bark path  around the outside  of the garden. Find the width  of the path and the outside dimensions.?

Solution:



Let x =width of the path
2x+10=total outside width
2x+20=total outside length
10*20=200 m^2= Inner area
216 m^2= Border Area (pine bark)

Equation:

Total Area= Inner area + Border Area
(2x+10)(2x+20)=200+216
4x^2+40x+20x+200=200+216
4x^2+60x=216
x^2+15x=54
x^2+15x-54=0
x^2+18x-3x-54=0
x(x+18)-3(x+18)=0
(x-3)(x+18)=0
Ignore (x+18)=0
x=-18
consider x-3=0
x=3 m width of the path
2x+10=2(3)+10=16 m total outside width
2x+20=2(3)+20=26 m Total outside length


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Question: The combined area of an 8" by 10" picture and its frame (border around the outside) is 288 sq inches . find the width and dimension of the picture frame?

 Solution:










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Question: The  8" by 10" picture is surrounded by a matte (border) whose area is 40 sq inches. (That is , the area of the matte is 40 sq inches).Find the width of the matte and the outside  dimensions of the matte?

 Solution:

 Let x be the width of the matte.



2x+8=total outside width of the frame
2x+10=total outside length of the frame.
Inner area=8*10=80 sq inches
Border Area=40 sq inches.

Total Area=Inner Area+ Border Area\
(2x+8)*(2x+10)=80+40
4x^2+16x+20x+80=80+40
4x^2+36x-40=0
x^2+9x-10=0
x^2+10x-x-10=0
x(x+10)-1(x+10)=0
(x-1)(x+10)=0
Therefore x=1 inch width of the frame.
The dimensions
2x+8=10 inches

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Question: A Cone has a base radius that is twice the height. If the volume of the cone is 36π cubic meters, find the dimensions of the cone.

 Solution: 

Let x be the height of the cone.
then radius r=2x
Given volume of the cone = 36π

The volume of cone V=1/3πr^2 h
36π=1/3π(2x)^2*x
36π=1/3*π*4x^2*x
x^3=27
x=3 meters of height
2*x=6 meters of radius.

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Question: A cylinder has a diameter that is three times the height.If the volume of the cylinder is 144π cubic inches, find the dimensions of the cylinder.

 Solution:

 Let x be the height of the cylinder.
Given that diameter is 3*x.
We know that radius =diameter/2
r=3/2*x
Given volume =144π cubic inches

Volume of the cylinder V=πr^2h
144π=π(3/2x)^2*x
144=9/4*x^3
64=x^3
x^3=64
x=4 inches height of the cylinder.
r=3/2*4=6 inches height of the cylinder.

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Question:  A cone has a base radius that is three times the height. If the volume of the cone is 375π cubic centimeters, find the dimension of the cone.

 Solution: 

Let x be the height of the cone.
then radius r=3*x
Given volume of the cone V=375π
The volume of the cone V=1/3πr^2h
375π=1/3*π*(3x)^2*x
375π=1/3*π*9x^3
375=3x^3
x^3=375/3
x^3=125*3/3
x^3=125
x=5 centimeters of height
r=3*x=15 centimeters of the radius.

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Question: The height of the cone is twice the radius of the base. If the volume of the cone is 144π cubic feet, find the dimension of the cone.

 Solution: 

Let x be the radius of the cone.
then height h=2*x
Given volume V=144π
Then Volume of the cone V=1/3πr^2h
144π=1/3πx^2*2x
144π=2/3π*x^3
x^3=216
x=6 feet of radius
h=2*x=12 feet of height.

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 Question: Find the radius of the sphere whose volume is 36π cubic centimeters?

 Solution: 

  Let x be the radius of the sphere.
then let us assume height of the sphere is equal to the radius say =x

Given Volume of the sphere is V= 36π

Then the volume of the sphere V=4/3πr^2*h

36π=4/3π*x^2*x
x^3=27
x=3 centimeters of the radius.

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Question: Find the radius of the sphere whose volume is 4.5π cubic centimeters?

 Solution: 

 Let x be the radius of the sphere.
then let us assume height of the sphere is equal to the radius say h =x

Given Volume of the sphere is V= 4.5π

Then the volume of the sphere V=4/3πr^2*h

Let x be the radius of the sphere.
then let us assume height of the sphere is equal to the radius say =x

Given Volume of the sphere is V= 36π

Then the volume of the sphere V=4/3πr^2*h

4.5π=4/3π*x^2*x
4.5=4/3*x^3
multiply by 10 on both sides
45=40/3*x^3
9=8/3*x^3
x^3=27/8
x=3/2 or 1.5 centimeters of radius 

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Question: A man want to build a rectangular speaker box of volume 3 cubic feet  for his 15 inch (diameter) of speakers. if the square base is 18 inches on each side , how deep are square boxes?

Solution:

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