Three-Dimensional Geometry: Volume and Surface Area Alıştırma Soruları — GRE General Test

Soru 1Soru →

A right circular cylinder has base radius $r$ and height $h$ . The height of the cylinder is increased by $50\%$ , and its base radius is decreased by $20\%$ . Which of the following statements about the modified cylinder compared to the original cylinder must be true? Select all that apply.

Geçerli olan tümünü seçin

The volume of the cylinder decreases by

4\%

.

The lateral surface area of the cylinder increases by

20\%

.

The combined area of the two circular bases decreases by

20\%

.

If

h = r

, the total surface area of the cylinder decreases.

If

h = 4r

, the total surface area of the cylinder decreases.

Cevabı ve açıklamayı göster

Cevap: The volume of the cylinder decreases by $4\%$ .; The lateral surface area of the cylinder increases by $20\%$ .; If $h = r$ , the total surface area of the cylinder decreases.

Cevap

The statements confirming a

4\%

volume decrease, a

20\%

lateral surface area increase, and a total surface area decrease when

h = r

are all true.

The volume of the modified cylinder decreases by

4\%

because

0.8^2 \times 1.5 = 0.96

. The lateral surface area increases by

20\%

because

0.8 \times 1.5 = 1.20

. When

h = r

, the original total surface area

4\pi r^2

reduces to

3.68\pi r^2

, confirming a decrease.

Adım Adım Çözüm

1

Express original and modified dimensions algebraically.

Original: radius

= r

, height

= h

. Modified: radius

= 0.8r

, height

= 1.5h

.

Decreasing radius by

20\%

multiplies it by

1 - 0.20 = 0.80

, while increasing height by

50\%

multiplies it by

1 + 0.50 = 1.50

.

2

Calculate the ratio of modified volume to original volume.

V_{new} = \pi (0.8r)^2 (1.5h) = 0.64 \times 1.5 \pi r^2 h = 0.96 V_{old}

, indicating a

4\%

volume decrease.

Volume of a cylinder is given by

V = \pi r^2 h

.

3

Calculate the ratio of modified lateral surface area to original lateral surface area.

L_{new} = 2\pi (0.8r)(1.5h) = 2.4\pi r h = 1.20 L_{old}

, indicating a

20\%

lateral surface area increase.

Lateral surface area of a cylinder is given by

L = 2\pi r h

.

4

Evaluate the base area change and total surface area under specific height-to-radius ratios.

Base area becomes

(0.8)^2 = 0.64

of original (

36\%

decrease). For

h = r

, total surface area changes from

4\pi r^2

to

3.68\pi r^2

(decrease). For

h = 4r

, total surface area changes from

10\pi r^2

to

10.88\pi r^2

(increase).

Total surface area is the sum of lateral surface area and base area,

A = 2\pi r^2 + 2\pi r h

.

Anahtar Kavram

Impact of dimensional scaling on volume, lateral surface area, and total surface area of cylinders

Soru 2Soru →

A solid right circular cylinder has a base radius of $r$ and a height of $4r$ . A solid sphere has a radius of $R$ . If the total surface area of the sphere is equal to the total surface area of the cylinder, what is the ratio of the volume of the sphere to the volume of the cylinder?

\frac{5\sqrt{10}}{12}

\frac{\sqrt{10}}{2}

\frac{2\sqrt{2}}{3}

\frac{5\sqrt{10}}{6}

\frac{5}{3}

Cevabı ve açıklamayı göster

Cevap: $\frac{5\sqrt{10}}{12}$

Cevap

The ratio of the volume of the sphere to the volume of the cylinder is

\frac{5\sqrt{10}}{12}

.

The total surface area of the cylinder is the sum of its lateral area and two circular bases:

2\pi r(4r) + 2\pi r^2 = 10\pi r^2

. Setting this equal to the sphere's surface area

4\pi R^2

yields

R/r = \sqrt{5/2} = \sqrt{10}/2

. The ratio of the sphere's volume

\frac{4}{3}\pi R^3

to the cylinder's volume

\pi r^2 (4r) = 4\pi r^3

is

\frac{1}{3}(R/r)^3 = \frac{1}{3} \left(\frac{\sqrt{10}}{2}\right)^3 = \frac{5\sqrt{10}}{12}

.

Adım Adım Çözüm

1

Calculate the total surface area of the cylinder.

\text{TSA}_{\text{cyl}} = 2\pi r^2 + 2\pi r h = 2\pi r^2 + 2\pi r(4r) = 10\pi r^2

A cylinder's total surface area consists of two circular bases (

2\times\pi r^2

) and the lateral surface area (

2\pi r h

).

2

Equate the total surface area of the sphere to the total surface area of the cylinder to find the ratio of

R

to

r

.

4\pi R^2 = 10\pi r^2 \implies R^2 = \frac{5}{2}r^2 \implies \frac{R}{r} = \sqrt{\frac{5}{2}} = \frac{\sqrt{10}}{2}

The total surface area of a sphere of radius

R

is

4\pi R^2

.

3

Express the volumes of both figures in terms of

r

and

R

.

V_{\text{sph}} = \frac{4}{3}\pi R^3

and

V_{\text{cyl}} = \pi r^2 h = \pi r^2 (4r) = 4\pi r^3

The volume of a sphere is

\frac{4}{3}\pi R^3

and the volume of a cylinder is

\pi r^2 h

.

4

Compute the ratio of the volume of the sphere to the volume of the cylinder.

\frac{V_{\text{sph}}}{V_{\text{cyl}}} = \frac{\frac{4}{3}\pi R^3}{4\pi r^3} = \frac{1}{3}\left(\frac{R}{r}\right)^3 = \frac{1}{3}\left(\frac{\sqrt{10}}{2}\right)^3 = \frac{1}{3} \cdot \frac{10\sqrt{10}}{8} = \frac{5\sqrt{10}}{12}

Substitute

\frac{R}{r} = \frac{\sqrt{10}}{2}

into the ratio expression.

Anahtar Kavram

Relating 3D surface area formulas to volume formulas for cylinders and spheres

Tahmini Süre:2m 0s

Soru 3Soru →

A right circular cylinder has a base radius of $3$ units and a height of $4$ units. Which of the following statements regarding this cylinder are true? Select all such statements.

Geçerli olan tümünü seçin

The volume of the cylinder is

36\pi

cubic units.

The total surface area of the cylinder is

42\pi

square units.

The volume of the cylinder is

24\pi

cubic units.

The total surface area of the cylinder is

36\pi

square units.

The ratio of the area of one circular base to the lateral surface area is

1

to

2

.

Cevabı ve açıklamayı göster

Cevap: The volume of the cylinder is $36\pi$ cubic units.; The total surface area of the cylinder is $42\pi$ square units.

Cevap

The correct statements are that the volume of the cylinder is

36\pi

cubic units and the total surface area of the cylinder is

42\pi

square units.

The volume of a cylinder with radius

3

and height

4

is

V = \pi r^2 h = \pi(3)^2(4) = 36\pi

. The total surface area is

A = 2\pi r h + 2\pi r^2 = 2\pi(3)(4) + 2\pi(3)^2 = 24\pi + 18\pi = 42\pi

. Therefore, both statements asserting these exact values are correct.

Adım Adım Çözüm

1

Calculate the volume of the right circular cylinder

Volume

V = \pi r^2 h = \pi (3^2)(4) = 36\pi

The formula for the volume of a right circular cylinder is

V = \pi r^2 h

.

2

Calculate the lateral surface area and total surface area of the cylinder

Lateral Area =

2\pi r h = 24\pi

; Total Area =

24\pi + 2(\pi \cdot 3^2) = 42\pi

Total surface area is the sum of the lateral surface area (

2\pi r h

) and the areas of the two circular bases (

2\pi r^2

).

3

Evaluate the base-to-lateral surface area ratio

Ratio =

\frac{9\pi}{24\pi} = \frac{3}{8}

Comparing the base area (

9\pi

) to the lateral area (

24\pi

) simplifies to

3:8

.

Anahtar Kavram

Volume and surface area formulas for right circular cylinders

Soru 4Soru →

A solid rectangular box has a length of $6$ centimeters, a width of $4$ centimeters, and a height of $5$ centimeters. What is the total surface area, in square centimeters, of the box?

74

120

148

240

288

Cevabı ve açıklamayı göster

Cevap: $148$

Cevap

The total surface area of the rectangular box is

148

square centimeters.

The total surface area of a rectangular solid with length

l

, width

w

, and height

h

is given by

2(lw + lh + wh)

. Substituting

l = 6

,

w = 4

, and

h = 5

yields

2(6 \cdot 4 + 6 \cdot 5 + 4 \cdot 5) = 2(24 + 30 + 20) = 2(74) = 148

square centimeters.

Adım Adım Çözüm

1

Identify the given dimensions of the rectangular box.

Length

l = 6

cm, width

w = 4

cm, and height

h = 5

cm.

These dimensions are required to compute the area of each face.

2

Calculate the surface area of the three distinct face pairs.

Top and bottom faces:

6 \times 4 = 24

; Front and back faces:

6 \times 5 = 30

; Left and right side faces:

4 \times 5 = 20

.

A rectangular solid has six rectangular faces grouped into three identical pairs.

3

Apply the surface area formula

A = 2(lw + lh + wh)

.

A = 2(24 + 30 + 20) = 2(74) = 148

square centimeters.

Summing the area of all six faces gives the total surface area.

Anahtar Kavram

Surface Area of a Rectangular Solid

Soru 5Soru →

A solid right circular cone has a base radius of $r$ and a height of $h = \frac{4}{3}r$ . A solid sphere has a radius of $R$ . If the total surface area of the cone is equal to the total surface area of the sphere, which of the following statements must be true? Select all that apply.

Geçerli olan tümünü seçin

The base radius of the cone,

r

, is strictly greater than the radius of the sphere,

R

.

The volume of the cone is strictly less than the volume of the sphere.

The ratio of the volume of the cone to the volume of the sphere is

\frac{\sqrt{6}}{4}

.

The volume of the cone is strictly greater than the volume of the sphere.

The ratio of the base radius of the cone to the radius of the sphere,

\frac{r}{R}

, is equal to

\frac{3}{2}

.

Cevabı ve açıklamayı göster

Cevap: The base radius of the cone, $r$ , is strictly greater than the radius of the sphere, $R$ .; The volume of the cone is strictly less than the volume of the sphere.; The ratio of the volume of the cone to the volume of the sphere is $\frac{\sqrt{6}}{4}$ .

Cevap

The statements confirming that the base radius of the cone is strictly greater than the radius of the sphere, that the volume of the cone is strictly less than the volume of the sphere, and that the ratio of the volume of the cone to the volume of the sphere is

\frac{\sqrt{6}}{4}

are all correct.

Equating the total surface area of the cone

\frac{8}{3}\pi r^2

with the surface area of the sphere

4\pi R^2

yields

\frac{r^2}{R^2} = \frac{3}{2}

, which simplifies to

\frac{r}{R} = \sqrt{\frac{3}{2}}

. Because

\sqrt{1.5} > 1

, the base radius of the cone is strictly greater than the radius of the sphere. Furthermore, evaluating the ratio of their volumes gives

\frac{V_{\text{cone}}}{V_{\text{sphere}}} = \frac{\frac{4}{9}\pi r^3}{\frac{4}{3}\pi R^3} = \frac{1}{3}\left(\frac{r}{R}\right)^3 = \frac{\sqrt{6}}{4} \approx 0.612

. Because this ratio is strictly less than

1

, the volume of the cone is strictly less than the volume of the sphere. Thus, the three true statements are those stating

r > R

, that the volume of the cone is strictly less than the sphere's volume, and that their volume ratio is

\frac{\sqrt{6}}{4}

.

Adım Adım Çözüm

1

Calculate the slant height and total surface area of the cone in terms of

r

.

Slant height

l = \sqrt{r^2 + \left(\frac{4}{3}r\right)^2} = \sqrt{\frac{25}{9}r^2} = \frac{5}{3}r

. Total surface area

A_{\text{cone}} = \pi r^2 + \pi r l = \pi r^2 + \pi r\left(\frac{5}{3}r\right) = \frac{8}{3}\pi r^2

.

The total surface area of a right circular cone is the sum of its base area

\pi r^2

and lateral area

\pi r l

.

2

Equate the total surface area of the cone to the total surface area of the sphere to find the ratio

\frac{r}{R}

.

\frac{8}{3}\pi r^2 = 4\pi R^2 \implies 2 r^2 = 3 R^2 \implies \frac{r^2}{R^2} = \frac{3}{2} \implies \frac{r}{R} = \sqrt{\frac{3}{2}} = \frac{\sqrt{6}}{2}.

The surface area of a sphere of radius

R

is

4\pi R^2

.

3

Compare the linear dimensions

r

and

R

.

Since

\frac{r}{R} = \sqrt{1.5} \approx 1.225 > 1

, it follows that

r > R

.

A ratio greater than

1

implies the numerator is larger than the denominator.

4

Express the volumes of both solids and compute their ratio

\frac{V_{\text{cone}}}{V_{\text{sphere}}}

.

V_{\text{cone}} = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi r^2 \left(\frac{4}{3}r\right) = \frac{4}{9}\pi r^3

.

V_{\text{sphere}} = \frac{4}{3}\pi R^3

. Therefore,

\frac{V_{\text{cone}}}{V_{\text{sphere}}} = \frac{\frac{4}{9}\pi r^3}{\frac{4}{3}\pi R^3} = \frac{1}{3}\left(\frac{r}{R}\right)^3 = \frac{1}{3}\left(\frac{3}{2}\right)^{3/2} = \frac{1}{3} \cdot \frac{3\sqrt{3}}{2\sqrt{2}} = \frac{\sqrt{6}}{4}

.

Using standard volume formulas for cones and spheres and substituting the known linear dimension ratio.

5

Determine whether the cone's volume is greater than or less than the sphere's volume.

Since

\frac{\sqrt{6}}{4} \approx 0.612 < 1

,

V_{\text{cone}} < V_{\text{sphere}}

.

A volume ratio less than

1

proves the cone has a smaller volume than the sphere.

Anahtar Kavram

Analyzing geometric scaling, volume, and total surface area relations between cones and spheres using variable constraints.

Soru 6Soru →

A solid sphere of radius $r$ is placed inside a hollow right circular cylinder of base radius $r$ and height $3r$ such that the sphere rests on the bottom base of the cylinder. Water is then poured into the cylinder until the water level just covers the top of the sphere (a height of $2r$ ). What is the ratio of the volume of the empty space remaining in the cylinder above the water level to the volume of the water inside the cylinder?

\frac{1}{3}

\frac{1}{2}

\frac{3}{4}

\frac{3}{2}

2

Cevabı ve açıklamayı göster

Cevap: $\frac{3}{2}$

Cevap

The ratio of the volume of the empty space remaining in the cylinder above the water level to the volume of the water inside the cylinder is

\frac{3}{2}

.

The empty cylinder portion at the top has height

r

, giving a volume of

\pi r^3

. The submerged section up to height

2r

has total volume

2\pi r^3

, of which the sphere accounts for

\frac{4}{3}\pi r^3

. Subtracting the sphere's volume leaves

\frac{2}{3}\pi r^3

of water. The ratio of empty space to water volume is

\frac{\pi r^3}{\frac{2}{3}\pi r^3} = \frac{3}{2}

.

Adım Adım Çözüm

1

Calculate the volume of the empty cylindrical space above the water height.

The total height of the cylinder is

3r

and the water level reaches height

2r

, leaving an empty height of

3r - 2r = r

. The volume of empty space is

V_{\text{empty}} = \pi r^2 \cdot r = \pi r^3

.

The empty space forms a smaller cylinder of base radius

r

and height

r

at the top.

2

Calculate the total volume of the cylindrical region filled up to the water height of

2r

.

V_{\text{region}} = \pi r^2 \cdot (2r) = 2\pi r^3

.

This represents the combined volume of the water and the submerged sphere.

3

Calculate the volume of the solid sphere of radius

r

.

V_{\text{sphere}} = \frac{4}{3}\pi r^3

.

Using the standard formula for the volume of a sphere,

V = \frac{4}{3}\pi r^3

.

4

Find the volume of the water by subtracting the sphere's volume from the total filled region.

V_{\text{water}} = 2\pi r^3 - \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3

.

The space taken up by the water is the total volume up to height

2r

minus the displacement volume of the sphere.

5

Compute the required ratio of

V_{\text{empty}}

to

V_{\text{water}}

.

\frac{V_{\text{empty}}}{V_{\text{water}}} = \frac{\pi r^3}{\frac{2}{3}\pi r^3} = \frac{1}{\frac{2}{3}} = \frac{3}{2}

.

Dividing the empty space volume by the water volume simplifies to

\frac{3}{2}

.

Anahtar Kavram

Displacement and volume calculations combining 3D geometric figures (cylinders and spheres).

Tahmini Süre:2m 30s

Soru 7Soru →

A hollow metallic spherical shell has an inner radius of $3$ centimeters and an outer radius of $r$ centimeters, where $r > 3$ . If the spherical shell is melted down and completely recast into a solid right circular cylinder with base radius $r$ centimeters and height $7$ centimeters, what is the value of $r$ ?

4.5

5.25

6

7.5

9

Cevabı ve açıklamayı göster

Cevap: $6$

Cevap

6

The volume of metal in the hollow spherical shell is

\frac{4}{3}\pi(r^3 - 3^3) = \frac{4}{3}\pi(r^3 - 27)

. The volume of the recast cylinder is

\pi r^2 h = 7\pi r^2

. Equating these volumes gives

\frac{4}{3}(r^3 - 27) = 7r^2

, which simplifies to

4r^3 - 21r^2 - 108 = 0

. Factoring this cubic equation gives

(r - 6)(4r^2 + 3r + 18) = 0

. Since the quadratic term has no real roots, the only real solution is

r = 6

.

Adım Adım Çözüm

1

Set up the formula for the volume of metal in the hollow spherical shell.

V_{\text{shell}} = \frac{4}{3}\pi \left(r^3 - 3^3\right) = \frac{4}{3}\pi \left(r^3 - 27\right)

The metal occupies only the region between the inner sphere of radius 3 cm and outer sphere of radius r cm.

2

Set up the formula for the volume of the recast solid right circular cylinder.

V_{\text{cylinder}} = \pi r^2 h = 7\pi r^2

The cylinder has base radius r cm and height 7 cm.

3

Equate the two volumes since no metal is lost during melting and recasting.

\frac{4}{3}\pi \left(r^3 - 27\right) = 7\pi r^2

Conservation of volume during recasting.

4

Simplify the equation and solve for r.

4(r^3 - 27) = 21r^2 \implies 4r^3 - 21r^2 - 108 = 0

Divide both sides by

\pi

and multiply by 3 to clear the fraction.

5

Factor the cubic polynomial

4r^3 - 21r^2 - 108 = 0

.

(r - 6)(4r^2 + 3r + 18) = 0 \implies r = 6

Testing

r = 6

gives

4(216) - 21(36) - 108 = 864 - 756 - 108 = 0

. The quadratic factor

4r^2 + 3r + 18

has a negative discriminant and produces no real roots.

Anahtar Kavram

Volume formulas for hollow spheres and right circular cylinders

Soru 8Soru →

Three adjacent faces of a right rectangular prism have surface areas of $24$ , $36$ , and $54$ square centimeters. What is the volume of the prism, in cubic centimeters?

Cevabı ve açıklamayı göster

Cevap: 216

Cevap

The volume of the rectangular prism is

216

cubic centimeters.

Let the edge lengths of the right rectangular prism be

l

,

w

, and

h

. The areas of three adjacent faces meeting at a common vertex are given by

lw = 24

,

wh = 36

, and

lh = 54

. Multiplying these three equations together yields

(lw)(wh)(lh) = (lwh)^2 = 24 \times 36 \times 54 = 46,656

. Since the volume

V

of a right rectangular prism is equal to

lwh

, taking the positive square root gives

V = \sqrt{46,656} = 216

cubic centimeters.

Adım Adım Çözüm

1

Define variables for the three dimensions of the rectangular prism.

Let

l

,

w

, and

h

represent the length, width, and height of the prism.

The area of each rectangular face is the product of two adjacent dimensions.

2

Express the given face areas in terms of the dimensions.

lw = 24

,

wh = 36

, and

lh = 54

.

Adjacent faces share pairwise dimensions.

3

Multiply all three face area equations together.

(lw)(wh)(lh) = l^2 w^2 h^2 = (lwh)^2 = 24 \times 36 \times 54 = 46,656

.

Multiplying the three face areas yields the square of the prism's volume (

V = lwh

).

4

Calculate the square root of the product to determine the volume.

V = lwh = \sqrt{46,656} = 216

.

Taking the square root gives the volume directly without needing to solve for individual edge lengths.

Anahtar Kavram

For a right rectangular prism with adjacent face areas

A_1

,

A_2

, and

A_3

, the volume is given by

V = \sqrt{A_1 \times A_2 \times A_3}

.

Tahmini Süre:1m 15s

Soru 9Soru →

A solid cube has a total surface area of $S$ and a volume of $V$ . A solid right circular cylinder has a height equal to its base diameter. If the total surface area of the cylinder is also equal to $S$ , what is the volume of the cylinder in terms of $V$ ?

\frac{2}{\sqrt{\pi}} V

\frac{\sqrt{\pi}}{2} V

\frac{1}{\sqrt{\pi}} V

\sqrt{\frac{27}{2\pi}} V

\frac{2}{\pi} V

Cevabı ve açıklamayı göster

Cevap: $\frac{2}{\sqrt{\pi}} V$

Cevap

\frac{2}{\sqrt{\pi}} V

The total surface area of a cube with edge

a

is

6a^2

and its volume is

V = a^3

. For a cylinder with base radius

r

and height

h = 2r

, the total surface area is

2\pi r^2 + 2\pi r(2r) = 6\pi r^2

. Setting

6\pi r^2 = 6a^2

yields

r = \frac{a}{\sqrt{\pi}}

and

h = \frac{2a}{\sqrt{\pi}}

. Substituting these into the volume formula

V_{\text{cyl}} = \pi r^2 h

produces

\pi \left(\frac{a^2}{\pi}\right) \left(\frac{2a}{\sqrt{\pi}}\right) = \frac{2a^3}{\sqrt{\pi}} = \frac{2}{\sqrt{\pi}} V

.

Adım Adım Çözüm

1

Express the surface area and volume of the cube in terms of its side length

a

.

Surface area

S = 6a^2

and volume

V = a^3

.

A cube with edge length

a

has

6

identical square faces of area

a^2

and volume

a^3

.

2

Set up the total surface area formula for the cylinder with radius

r

and height

h = 2r

, and equate it to

S

.

Total surface area

S_{\text{cyl}} = 2\pi r^2 + 2\pi r h = 2\pi r^2 + 2\pi r (2r) = 6\pi r^2 = 6a^2

.

The cylinder's height is equal to its base diameter (

2r

). Equating surface areas gives

6\pi r^2 = 6a^2

.

3

Solve for radius

r

in terms of edge length

a

.

r^2 = \frac{a^2}{\pi} \implies r = \frac{a}{\sqrt{\pi}}

.

Dividing both sides by

6\pi

and taking the square root isolates

r

.

4

Calculate the volume of the cylinder in terms of

V

.

V_{\text{cyl}} = \pi r^2 h = \pi \left(\frac{a^2}{\pi}\right) \left(\frac{2a}{\sqrt{\pi}}\right) = \frac{2a^3}{\sqrt{\pi}} = \frac{2}{\sqrt{\pi}} V

.

Substituting

r^2 = \frac{a^2}{\pi}

and

h = \frac{2a}{\sqrt{\pi}}

into the cylinder volume formula

\pi r^2 h

yields the answer in terms of

V = a^3

.

Anahtar Kavram

Volume and surface area relationship between geometric solids

Tahmini Süre:2m 0s

Soru 10Soru →

A right circular cylinder has a base radius of $r$ and a height of $h$ . If the base radius is doubled and the height is halved to create a new right circular cylinder, which of the following statements must be true? Select all that apply.

Geçerli olan tümünü seçin

The volume of the new cylinder is twice the volume of the original cylinder.

The lateral surface area of the new cylinder is equal to the lateral surface area of the original cylinder.

The total surface area of the new cylinder is twice the total surface area of the original cylinder.

The volume of the new cylinder is four times the volume of the original cylinder.

The area of the circular base of the new cylinder is four times the area of the circular base of the original cylinder.

Cevabı ve açıklamayı göster

Cevap: The volume of the new cylinder is twice the volume of the original cylinder.; The lateral surface area of the new cylinder is equal to the lateral surface area of the original cylinder.; The area of the circular base of the new cylinder is four times the area of the circular base of the original cylinder.

Cevap

The correct statements are: the volume of the new cylinder is twice the volume of the original cylinder, the lateral surface area of the new cylinder is equal to the lateral surface area of the original cylinder, and the area of the circular base of the new cylinder is four times the area of the circular base of the original cylinder.

The volume formula

V = \pi r^2 h

scales by

2^2 \times \frac{1}{2} = 2

, so volume doubles. The lateral surface area formula

L = 2\pi r h

scales by

2 \times \frac{1}{2} = 1

, remaining unchanged. The base area formula

A = \pi r^2

scales by

2^2 = 4

, quadrupling the area.

Adım Adım Çözüm

1

Write down the original formulas for a cylinder with radius

r

and height

h

.

Base Area

A_1 = \pi r^2

, Lateral Surface Area

L_1 = 2\pi r h

, Total Surface Area

T_1 = 2\pi r h + 2\pi r^2

, Volume

V_1 = \pi r^2 h

.

Establish baseline geometric properties.

2

Substitute the new dimensions

r' = 2r

and

h' = \frac{h}{2}

into each formula.

New Base Area

A_2 = \pi (2r)^2 = 4\pi r^2 = 4A_1

.
New Lateral Surface Area

L_2 = 2\pi (2r) \left(\frac{h}{2}\right) = 2\pi r h = L_1

.
New Volume

V_2 = \pi (2r)^2 \left(\frac{h}{2}\right) = \pi (4r^2) \left(\frac{h}{2}\right) = 2\pi r^2 h = 2V_1

.

Calculate the updated values for each geometric property.

3

Evaluate each statement against the calculated values.

Volume doubles (

V_2 = 2V_1

), lateral surface area remains equal (

L_2 = L_1

), and base area quadruples (

A_2 = 4A_1

).

Determine which options state correct mathematical relationships.

Anahtar Kavram

Scaling effects on 3D geometry formulas for right circular cylinders

Three-Dimensional Geometry: Volume and Surface Area

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