Volume of Pyramids and Cones Worksheet

**Page 1**

1.

3 iron bars of dimensions 1.25 m × 1.25 m × 5 m are melted to mould conical pieces of base radius 10 cm and height 20 cm. Find out the number of pieces that can be made taking $\pi $ = 3.

a. | 3906 | ||

b. | 85 | ||

c. | 117 | ||

d. | 11719 |

[Formula.

Volume of the iron bars = 3(1.25 × 1.25 × 5) = 23.438 m³

[Simplify.]

Volume of one cone =

[Volume of a cone =

Number of cones that can be moulded =

[1 m

[Substitute and simplify.]

Correct answer : (4)

2.

A pile of sand is in the form of cone. The circumference of its base is measured as 540 cm. The slant height is measured as 150 cm. This has to be transported in a lorry. The sand is loaded on to the lorry at the rate of 4500 c.c/min. How long will it take to load the entire sand on the lorry? [Take $\pi $ = 3.]

a. | 224 min | ||

b. | 221 min | ||

c. | 222 min | ||

d. | 216 min |

[Given.]

Radius of the cone (r) =

[Radius of the cone (r) =

Slant height = 150 cm

[Given.]

Height of the cone(h) =

[Height of the cone =

Volume of the sand =

[Volume of the cone =

Rate of loading the sand = 4500 c.c/min.

[Given.]

Time taken to load =

[Formula.]

Time taken to load =

[Substitute in step 7 and simplify.]

Correct answer : (4)

3.

121 litres of grains is stored on a paddy field in the form of a cone. The height of the vertex of the cone from the ground is 250 cm. Find the circumference of the base of the cone. [Take $\pi $ = 3. ]

a. | 137 cm | ||

b. | 132 cm | ||

c. | 157 cm | ||

d. | 144 cm |

[Given.]

[1 litre = 1000 c.c .]

Height of the cone = 250 cm

[Given.]

[Volume of a cone =

[Divide each side by 250.]

[Simplify.]

Circumference of the base of the cone = 2 × 3 × 22 = 132 cm

[Circumference = 2

Correct answer : (2)

4.

A metallic solid in the form of a square pyramid of base 16 cm and height 2 cm is melted to make conic pieces of base radius 3 cm and height 3 cm. How many pieces can be made approximately? [Take $\pi $ = 3.]

a. | 35 | ||

b. | 25 | ||

c. | 28 | ||

d. | 6 |

[Formula.]

Volume of the pyramid =

[Volume of pyramid =

Volume of one cone =

[Volume of one cone =

Number of cones that can be made =

[Substitute in step 1.]

Correct answer : (4)

5.

A tent is in the form of a square pyramid with a base length of 21 m. The slant height of the pyramid is 16 m. If the rate of making the tent is $60 per cubicmeter, then what is the cost of making the tent?

a. | $106496.40 | ||

b. | $106507.40 | ||

c. | $106457.40 | ||

d. | $106485.40 |

[Given.]

Slant height = 16 m.

Height of the tent =

[Height of tent =

Volume of the tent =

[Volume of the pyramid =

Rate of making = $60 per m

[Given.]

Cost of making the tent = Volume of the tent × Rate of making

[Formula.]

Cost of making the tent = 1774.29 × 60 = $106457.40

[Substitute in step 6 and simplify.]

Correct answer : (3)

6.

A cone has a base radius of $r$ and height $h$. If $x$ is the measure of the height of a similar cone with half the volume of the original cone, then what is the relation between $x$ and $h$?

a. | $h$ ^{3} = 2 $x$^{3} | ||

b. | $x$ = $\sqrt{\frac{2h}{3}}$ | ||

c. | $x$ = $\frac{h}{2}$ | ||

d. | $x$ = $\sqrt{h}$ |

AB =

[Δ ABO ~ Δ CDO.]

Volume of the original cone =

[Formula.]

Volume of the second cone =

[From the figure, height =

Volume of the second cone =

[From step 1, CD =

Volume of first cone = 2 (Volume of the second cone).

[According to the data.]

[From steps 2 and 4.]

[Simplify.]

The relation between

[Multiply each side by

Correct answer : (1)

7.

What is the ratio of volumes of a cone of radius 28 cm and height 42 cm to a cylinder of radius 7 cm and height 14 cm? [Take $\pi $ = 3.]

a. | 16 : 1 | ||

b. | 22 : 1 | ||

c. | 21 : 1 | ||

d. | 24 : 1 |

[Volume of the cone =

Volume of the cylinder = 3 × (7)

[Volume of the cylinder =

Ratio of the volumes =

[Formula.]

=

[From steps 1 and 2.]

Ratio of the volumes = 16 : 1

Correct answer : (1)

8.

A tank is in the shape of a cone with a depth of 20 cm and maximum inner radius of 30 cm. Water is pumped into it at a rate of 900 c.c per minute. How long it takes for the vessel to get filled up? [ Take $\pi $ = 3. ]

a. | 20 minutes | ||

b. | 74 minutes | ||

c. | 52 minutes | ||

d. | 36 minutes |

[Formula.]

Volume of the vessel =

[Volume of the cone =

Rate of pumping = 900 c.c per minute

[Given.]

Time taken to fill up the tank =

[From steps 2 and 3.]

Correct answer : (1)

9.

The height of 3 ounces of liquid in a conical cup is half of the height of the cup. How many ounces of the liquid will be required to fill the entire conical cup?

a. | 24 ounces | ||

b. | 6 ounces | ||

c. | 29 ounces | ||

d. | 12 ounces |

[Volume of the cone =

Volume of the part of the conical cup containing liquid in it,

Consider ΔABC and ΔDEC,

[As ΔABC and ΔDEC are similar triangles.]

[Given,

[Simplify.]

[Formula.]

[Volume of the liquid = 3 ounces.]

[From steps 4 and 5.]

V = 3 × 4 × 2

[Multiply with 3 on both sides.]

V = 24 ounces

[Simplify.]

So, the volume of the liquid required to fill the entire conical cup = 24 ounces

Correct answer : (1)

10.

A metallic cone of radius 8 cm and height 18 cm is melted and made into identical spheres each of radius 2 cm. How many spheres can be made?

a. | 36 | ||

b. | 24 | ||

c. | 48 | ||

d. | 72 |

[Formula.]

Volume of a cone =

[Volume of a cone =

Volume of each sphere =

[Volume of sphere =

Number of spheres =

[Substitute in step 1 and simplify.]

Therefore, 36 spheres can be made.

Correct answer : (1)