Question

A work bench is in the shape of a trapezoid, as shown in figure. if the perimeter of the workbench is 260 cm, what is its area if y greater than x?

Answer 1

Perimeter = x + x + y + 65 =     65 + 2x + y  = 260
                   2 x + y = 195    =>  x + (x+y) = 195 
                   x+y = 195 - x        --------------------    equation 1
                   y - x  =  195 - 3x        -----------------  equation 2

Draw a perpendicular from the left end of 65.0 cm side , on to the side of y cm.
We get  a square of size  x by x  and a right angle triangle, whose width is (y-x) and height is x.

Area of trapezoid =  area of square + area of triangle
                 = x²  +  1/2  x  ( y - x )  = x² + 1/2 x y - 1/2 x²
               = 1/2 x ( x + y )              substitute value of x+y from equation 1
             = 1/2 x [195 - x ]  = 97.5 x - 0.5 x²

IN the right side triangle,
x² + (y-x)² = 65²
x² + (195 - 3x)² = 65²
x² + 195² + 9x² - 2*3*195x = 65²
10 x² - 1170 x + 195² - 65² = 0    on simplification we get
x² - 117 x + 3380 = 0

Δ = 117²  - 4 * 3380 = 169

x = (117 + - 13 )/2  =  65 cm or 52 cm

Now y = 195 - 2x =    65 cm  for x= 65 cm ,   or  91 cm  for x = 52 cm

As it is said that  y > x  we choose the second set for x and y

Now area : 97.5 x - 0.5 x²  =  97.5 * 52 - 0.5 * 52² =  3718 cm²
Please see Diagram.