Question

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?

Answer 1

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`