Question 975313
<pre>
In the following I will use h, S and V respectively for height,
surface area and volume, respectively, with subscripts A,B,C,D,E
referring to the particular cones they are of.
</pre>
five cones A, B, C, D and E are mathematically similar.
their heights are in the ratio 2:3:4:5:6
<pre>

2:3:4:5:6 = 2k:3k:4k:5k:6k for any non-zero constant k 
</pre>
a) work out height of cone C.
<pre>
h<sub>A</sub>:h<sub>B</sub>:h<sub>C</sub>:h<sub>D</sub>:h<sub>E</sub> = 2k:3k:4k:5k:6k
for any positive k

Let's find the particular positive k such that 

h<sub>A</sub>=2k, h<sub>B</sub>=3k, h<sub>C</sub>=4k, h<sub>D</sub>=5k, h<sub>E</sub>=6k, 
</pre>
Cone B has height 3.6cm
<pre>
h<sub>B</sub> = 3k
 
3.6 = 3k
1.2 = k

Then with that constant k = 1.2, we can find any of the heights.
In particular,

h<sub>C</sub> = 4k = 4(1.2) = 4.8cm
</pre>
b) work out the surface area of cone D.
<pre>
Proportions of corresponding areas are the squares of the proportions of
corresponding linear measures.

S<sub>A</sub>:S<sub>B</sub>:S<sub>C</sub>:S<sub>D</sub>:S<sub>E</sub> = 2<sup>2</sup>:3<sup>2</sup>:4<sup>2</sup>:5<sup>2</sup>:6<sup>2</sup> = 4:9:16:25:36 = 4k:9k:16k:25k:36k 
for any positive k

Let's find the positive k such that 

S<sub>A</sub>=4k, S<sub>B</sub>=9k, S<sub>C</sub>=16k, S<sub>D</sub>=25k, S<sub>E</sub>=36k, 
</pre>
Cone B has...surface area 45cm^2. 
<pre>
S<sub>B</sub> = 9k
 
45 = 9k
5 = k

Then with that constant k=5, we can find any of the surface areas.
In particular,

S<sub>D</sub> = 25k = 25(5) = 125cm<sup>2</sup> 
</pre> 
c) Cone A is used to fill cone E with sand. how many full cones of sand 
from cone A are needed to fill cone E?
<pre>
Proportions of corresponding volumes are the cubes of the proportions of
corresponding linear measures.

V<sub>A</sub>:V<sub>B</sub>:V<sub>C</sub>:V<sub>D</sub>:V<sub>E</sub> = 2<sup>3</sup>:3<sup>3</sup>:4<sup>3</sup>:5<sup>3</sup>:6<sup>3</sup> = 8:27:64:125:216 = 8k:27k:64k:125k:216k
for any positive k

There exists positive k such that 

V<sub>A</sub>=8k, V<sub>B</sub>=27k, V<sub>C</sub>=64k, V<sub>D</sub>=125k, V<sub>E</sub>=216k, 

But we don't need to find that value of k as we did in the other two. We
simply divide V<sub>E</sub> = 216k by V<sub>A</sub>=8k, and the k's will cancel.

{{{(216k)/(8k)}}} = 27 full cones of sand from cone A are needed to fill cone E.

Edwin</pre>