It's Geometry

Question

A tepee with a dirt floor in the shape of a right cone has a slant height of 26 feet and a radius of 12.5 feet. Approximately how much canvas would be needed to cover the tepee?

 

The ratio between the volumes of two cubes is 125 to 216. What is the ratio between their respective surface areas?

 

Lea made two candles in the shape of right rectangular prisms. The first candle is 15 cm high, 8 cm long, and 8 cm wide. The second candle is 5 cm taller but has the same length and width. How much additional wax was needed to make the taller candle?

Answer 1

curved surface of a right cone is given by the formula:

pi*r*√(r²+h²) where r=radius of the base and h=altitude of the cone

find the altitude first

h²+12.5²=26²

h²+156.25=676

h²=676-156.25

h²=519.75

h=22.8 feet

3.14*12.5*√[(12.5)²+(22.8)²]

3.14*12.5*√(156.25+519.84)

3.14*12.5*√(676.09)

3.14*12.5*26 (notice that this is the slant height because h²+r²=(slant height)²)

finish from here...

 

 

125/216

the edges are 5 and 6 because 5³=125 and 6³=216

a cube is made up of 6 squares; find the area of one and multiply by 6

5*5=25, 6*25=150

6*6=36, 6*36=216

the ratio is 150/216=75/108=25/36 (all that was needed was 25 and 36 because each was multiplied by 6)

 

 

V=l*w*h

V=15*8*8=960 cm³

V=20*8*8=1280 cm³

1280-960=320 cm³ additional wax

Answer 2

1.  The surface area of a right cone is:

 

A = (pi)*radius*(slant height) + (pi)*(radius)²

 

The (pi)*(radius)² term is the area of the circular base of the cone.  Since it's a dirt floor, we don't have to cover the circular base with canvas.  So the area to be covered by canvas is just:

 

A = (pi)*radius*(slant height)

 

Plug in the values for the radius (12.5 ft), slant height (26 ft), and pi (3.14) to get the answer.

 

2. Volumes are proportional to the cube of the side length, surface area to the square of the side length.

 

125:216 = 5³:6³

 

So the ratio of the areas will be 5²:6² = 25:36

 

3. The volumes of the two candles are:

 

Volume (candle 1) = 15*8*8 = 960 cm³

 

Volume (candle 2) = (15+5)*8*8 = 20*8*8 = 1280 cm³

 

Subtract volume 1 from volume 2 to get the answer.