Question

A 5 m 60 cm high vertical pole casts a shadow 3 m 20 cm long. Find at the same time − (i) the length of the shadow cast by another pole 10 m 50 cm high
(ii) the height of a pole which casts a shadow 5 m long.

Answer 1

Hint: Here we will use the concept of direct proportion which states that if two quantities increase or decrease in the same ratio i.e. the ratio of increasing or decreasing remains constant then we say the quantities are directly proportional.
If x and y are in direct proportion then \[\dfrac{x}{y} = r\] where r is a constant.

Complete step-by-step answer:
We know that:-
1m = 100 cm
Therefore,
5 m 60 cm = \[\left( {5 \times 100} \right) + 60\]
5 m 60 cm = 560 cm
Similarly,
  3 m 20 cm = \[\left( {3 \times 100} \right) + 20\]
  3 m 20cm = 320 cm
  10 m 50 cm = \[\left( {10 \times 100} \right) + 50\]
  10 m 50 cm = 1050 cm

(i) Here let the length of the shadow cast by the pole of length 10 m 50 cm be x.
Now, since the height of the pole and length of the shadow are in direct proportion.
Therefore,
\[\dfrac{{560}}{{320}} = \dfrac{{1050}}{x}\]
Solving for the value of x we get:-
\[x = \dfrac{{1050 \times 320}}{{560}}\]
Solving it further we get:-
\[x = \dfrac{{1050 \times 320}}{{560}}\]
\[ \Rightarrow x = \dfrac{{1050 \times 4}}{7}\]
\[ \Rightarrow x = 600\] cm
Hence the length of the shadow of the other pole is 600 cm i.e. 6m.
(ii) Here, let us assume the height of the pole to be x.
Now we know that the height of the pole and the length of the shadow are in direct proportion.
Therefore, we get:-
\[\dfrac{{560}}{{320}} = \dfrac{x}{{500}}\]
Solving for x we get:-
\[x = \dfrac{{500 \times 560}}{{320}}\]
Solving it further we get:-
\[ \Rightarrow x = \dfrac{{500 \times 7}}{4}\]
\[ \Rightarrow x = 875\] cm
Hence the height of the another pole is in this case is 875 cm i.e. 8m 75cm

Note: Students should take a note that the height of the pole and the length of the shadow are in direct proportion because as the height of the pole increases the length of the shadow also increases.
The calculations should be done carefully to avoid mistakes in the answer.