School

Harvard UniversityDepartment

MathematicsCourse Code

Mathematics 21aProfessor

mathStudy Guide

MidtermThis

**preview**shows page 1. to view the full**5 pages of the document.**Notes: Important formulas to be kept in mind:

Parameters of a Cone: Radius of the base (r ), Height of the cone (h ) and Slant

Height of a Cone (l )

Volume of a Cone = \frac{1}{3} \pi r^2 h

Curved Surface area of a Cone = \pi r l

Total Surface area of a Cone = \pi r^2+ \pi r l

Question 1: Find the volume of a cone whose slant height is 17 \ cm and radius

of base is 8 \ cm .

Answer:

Volume of a Cone = \frac{1}{3} \pi r^2 h

l = 17 \ cm, \ r = 8 \ cm

Therefore h = \sqrt{l^2 - r^2} = \sqrt{17^2-8^2} = 15 \ cm

Therefore Volume = \frac{1}{3} \times \frac{22}{7} \times (8)^2 \times (15) =

1005.71 \ cm^3

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Question 2: The curved surface area of a cone is 12320 \ cm^2 . If the radius of

its base is 56 \ cm , find its height.

Answer:

Curved surface area of the cone = 12320 \ cm^2

r = 56 \ cm

\pi r l = 12320 \Rightarrow l = \frac{12320}{\pi . 56} = 70 \ cm

Therefore h = \sqrt{l^2 - r^2} = \sqrt{70^2-56^2} = 1542 \ cm

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Question 3: The circumference of the base of a 12 \ m high conical tent is 66 \

m . Find the volume of the air contained in it.

Answer:

h = 12 \ m

Circumference of the base = 66 \ m

Therefore 2 \pi r = 66 \Rightarrow r = \frac{33}{\pi}

Volume of a Cone = \frac{1}{3} \pi r^2 h

= \frac{1}{3} \times \pi \times (\frac{33}{\pi})^2 \times (12) = 1386 \ m^3

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Question 4: The radius and the height of a right circular cone are in the ratio

5:12 and its volume is 2512 \ cm^3 . Find the radius and slant height of the

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cone. (Take \pi = 3.14 )

Answer:

Given: Radius and the height of a right circular cone are in the ratio 5:12

Let r = 5x and h = 12 x

Volume = 2512 \ cm^3

Therefore: \frac{1}{3} \pi r^2 h = 2512

\Rightarrow \frac{1}{3} \times 3.14 \times (5x)^2 \times (12x) = 2512

\Rightarrow x^3 = 8 \ or \ x = 2

Therefore Radius = 5 (2) = 10 \ cm and Height = 12(2) = 24 \ cm

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Question 5: Two right circular cones X and, Y are made. X having three times

the radius of Y and Y having half the volume of X . Calculate the ratio

between the heights of X and Y .

Answer:

Let the radius of cone Y is x . Therefore the radius of cone X = 3x .

Let the height of cone Y = h_y and the height of cone X = h_x

Given: V_y = \frac{1}{2} V_x

\Rightarrow \frac{1}{3} \pi (x)^2 (h_y) = \frac{1}{2} \{ \frac{1}{3} \pi (3x)^2

(h_x) \}

\Rightarrow \frac{h_x}{h_y} = \frac{x^2}{(3x)^2}

\Rightarrow \frac{h_x}{h_y} = \frac{2}{9}

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Question 6: The diameters of two cones are equal. If their slant heights are in

the ratio 5 : 4 , find the ratio of their curved surface areas.

Answer:

Let the radius of Cone 1 and Cone 2 = x

Let the slant height of the cones be l_1 and l_2 respectively.

Curved Surface Area of a cone = \pi r l

Therefore

\frac{Curved \ Surface \ Area \ of \ Cone \ 1}{Curved \ Surface \ Area \ of \

Cone \ 2} = \frac{\pi \times x \times l_1}{\pi \times x \times l_2} = \frac{5}

{4}

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Question 7: There are two cones. The curved surface area of one is twice that of

the other. The slant height of the latter is twice that of the former. Find the

ratio of their radii.

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