Question

The rectangular front windshield of the bus has the length of 1.5√3 m and the width of 1.5 m . The wipers are attached to the bottom corners. If the length of each wiper is equal to the width of the windshield, find the percentage of the windshield that wipers can reach.
(Solve this task without using Trigonometry)​

Answer 1

givεη:-

The rectangular front windsheid of the bus has the length of 1.5√3 m and the width if 1.5 m

The wipers are attached to the bottoms of the corners..

The length of each wiper is equal to the width of windsheid..

тσ ғiη∂:-

The percentage of the windsheid that wipers can reach..

cση∂iтiση:-

Solve this task without using trigonometry

sσℓυтiση:-

вү υsiηg нεяση ғσямυℓα:

Area of Rectangle=Length×breadth

sυвsiтυ∂ε,

length = 1.5√3 and

breadth = 1.5 to find the area of windsheid..

Area of a windsheid=1.5√3×1.5=2.25√3=3.897

Area covered by wipers is area covered by sector AOD sector BOC and triangle AOB..

Area of a triangle AOD is using heron formula.

вү υsiηg нεяση ғσямυℓα:-

$\sqrt{s(s-a)(s-b)(s-c)}$

• s is for semi perimeter
• a,b,c are for sides of triangle

AO=1.5,BO=1.5,AB=1.5√3

Substituting values in formulas and simplify to find area of triangle AOB

=$\ \dfrac{(1.5)(1.5)√3}{4}$

Area of triangle AOB using formula$\dfrac{base \times height}{2}$

Equate both area to find OM =$\dfrac{1.5}{2}$

Hence O is the center of Rectangle.

Hence DO=AO= BO=CO (As Diagonals of rectangle are equal and bisect each other )

Triangle AOD and Triangle BOC are Equilateral triangle (As all sides are Equal)

Hence angle DAO is 60° and angle CBO is 60°

Area of Circular sector is:\dfrac${\text{sector angle}}{360} \pi (radius)^2$

Area of Circular sector AOD = BOC =$\dfrac{60}{360} \pi (1.5)^2$

$\dfrac{60}{360} \pi (1.5)^2=\dfrac{1}{6} \pi (2.25)$

Total area covered by wipers is : Area of Circular sector AOD + Area of Circular sector BOC + Area of Triangle AOB

$\dfrac{1}{6} \pi (2.25)+\dfrac{1}{6} \pi (2.25)+ \dfrac{(1.5)(1.5)\sqrt{3}}{4}$=3.33

the percentage of the windshield that wipers can reach is :

$\dfrac{3.33}{3.897} \times100 = 85.46 \% </p><p>3.897$

Another Method Using Pythagoras theorem :

AC² = AB² + BC²

=> AC² = (1.5√3)² + 1.5²

=> AC² = 1.5²(3 + 1)

=> AC = 1.5(2)

=> AC = 3

AO = 1.5 = $\dfrac{AC}{2}$

Hence O is the center of rectangle

=> AO = BO = CO = DO = 1.5