The perimeter of a trapezoid is #42 cm#; the oblique side is #10cm# and the difference between bases is #6 cm#. Calculate: a) The area b) Volume obtained by rotating the trapezoid around the base major?

1 Answer
Mar 22, 2018

drawn

Let us consider an isosceles trapezoid #ABCD# representing the situation of the given problem.

Its major base #CD=xcm#, minor base #AB=ycm#, oblique sides are #AD=BC=10cm#

Given #x-y=6cm.....[1]#

and perimeter #x+y+20=42cm#

#=>x+y=22cm.....[2]#

Adding [1] and [2] we get

#2x=28=>x=14 cm#

So #y =8cm#

Now #CD= DF=k=1/2(x-y)=1/2(14-8)=3cm#

Hence height #h=sqrt(10^2-k^2)=sqrt91cm#

So area of the trapezoid

#A=1/2(x+y)xxh=1/2xx(14+8)xxsqrt91=11sqrt91cm^2#

It is obvious that on rotating about major base a solid consisting of two similar cones in two sides and a cylinder at the middle will be formed as shown in above figure.

So total volume of the solid

#=2xx"volume of a cone" + "volume of a cylinder"#

#=[2xx1/3pi(sqrt91)^2xx3 + pixx(sqrt91)^2xx8]cm^3#

#=910picm^3#