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CBSE Class 10 Answered

Please solve the NCERT question No.4 Exercise-13.5 (p-258)
Asked by Mdivya | 12 Mar, 2009, 11:21: PM
Expert Answer
--------------------- +P --------------------
| /|\ |
| / | \ |
| / | \ S-s |H-h
| / | \ |
| / | \ |
| /*****|**r**\ |
| ** C+-----*+D--------------
H| S / *********** \ |
| / | \ |
| / | \ |
| / | \ s |
| / | \ |h
| / | \ |
| / | \ |
| / **********|********** \ |
| /***** | R *****\ |
-----* A+----------------+B----
****** ******
*********************

We know R, r, and h, but not H, the total height of the cone from
which the frustum was cut. If we can find it, then the volume of the
frustum will be the volume of the whole cone, pi R^2 H/3, minus the
volume of the cone we cut off the top, pi r^2 (H-h)/3.

The triangles PAB and PCD are similar, so we can write the equation

AB CD R r
-- = -- or - = ---
PA PC H H-h

Cross-multiplying [that is, multiplying both sides by H(H-h)], we get

R(H-h) = rH

We can distribute the left side and collect H terms, then divide:

RH - Rh = rH

RH - rH = Rh

(R-r)H = Rh

Rh
H = ---
R-r

Now let's write the volume formula and substitute this formula for H:

pi pi
V = -- R^2 H - -- r^2 (H-h)
3 3

pi
= -- (R^2 H - r^2 H + r^2 h)
3

pi
= -- [(R^2 - r^2) H + r^2 h]
3

pi Rh
= -- [(R^2 - r^2) --- + r^2 h]
3 R-r

pi R
= -- [(R^2 - r^2) --- + r^2] h
3 R-r

We can write R^2 - r^2 as (R - r)(R + r) and cancel:

pi
= --- [(R + r) R + r^2] h
3

pi
= --- [R^2 + Rr + r^2] h
3

That's the formula.
Answered by | 13 Mar, 2009, 01:55: AM
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