  Question

# A carpenter makes stools for electricians with a square top of side 0.5m and at a height 1.5m above the ground. Also, each leg is inclined at an angle of 60∘ to the ground. The lengths of two steps in metres correct up to two places of decimals is  1.045, 1.782 m 1.122, 1.901 m 1.077, 1.653 m 1.324, 1.890 m

Solution

## The correct option is C 1.077, 1.653 m As shown in the figure, AX and BY are the two legs at an angle of 60∘ to the ground. CD and EF are two steps. AR and BS are perpendiculars on the base XY. AR meets CD at L and EF at P. BS meets CD at M and EF at Q. We have, AR = BS = 1.5 m, AB = 0.5 m AL = LP = PR = BM = MQ = QS = 0.5 m From right angled ΔAXR, sin 60∘=ARAX⇒√32=1.5AX⇒ AX=1.5×2√3=3√3 ⇒AX=√3 m⇒ AX=1.73 m⇒AX=BY=1.73 m In right angled ΔACL and ΔAEP, ∠ACL=60∘ and ∠AEP=60∘ Then ALCL=tan 60∘ and APEP=tan 60∘⇒ 0.5CL=√3 and 1EP=√3⇒ CL=0.5√3 and EP=1√3⇒ CL=0.5×√33 and EP=√33 Similarly, MD=0.5×√33 and QF=√33 Now, CD = CL + LM + MD =0.5×√33+0.5+0.5×√33=0.5+√33=0.5+1.733 = 0.5 + 0.577 (approx) = 1.077 m Again, EF=EP+PQ+QF=√33+0.5+√33=0.5+2√33=0.5+2×1.733=0.5+3.463=0.5+1.153=1.653m Hence, and length of step CD = 1.077 m and length of step EF = 1.653 m  Suggest corrections   