Question

To find the area of a trapezoid, Dylan uses the formula a= 1/2(b1 + b2)h The bases have lengths of 3.6 cm and 12 1/3 cm. The height of the trapezoid is the square root of 5 cm. The area of the trapezoid is irrational because

Answer 1

Because the irrational number can not be expressed as the ratio of two integers and it is not an imaginary number.

Explanation:

Given

The length of one base of trapezoid = 3.6cm =

The length of other base of trapezoid =
`12(1)/(3)`=
`(37)/(3)`

The height of trapezoid=
`√(5)` cm

By using formula of area of trapezoid, a=
`(1)/(2) (b_1+b_2)* h`

Substitute the value of
`b_1, b_2` and h in the formula of area of trapezoid

Area of trapezoid=
`(1)/(2) ((36)/(10) +(37)/(3) )* √(5)`

Area of trapezoid=
`(1)/(2) ((108+370)/(30)) * √(5)`

Area of trapezoid=
`(1)/(2) * (478)/(30) *√(5)`

Area of trapezoid=
`(239)/(30) √(5)`
`cm^2`

Let a and b are two integers and let area of trapezoid is rational number

Therefore, the rational number can be write in the ratio of two integers a and b

`\therefore (239)/(30) √(5)=(a)/(b)` where
`b \\eq 0`

`(30a)/(239b) =√(5)`

We know that
`√(5)` is a irrational number and
`(30a)/(239b)` is a rational number

Irrational number can never be equal to rational number

Therefore, the area of trapezoid is irrational number.

Answer 2

b1 + b2 =

`(36)/(10) + (37)/(3) = (108)/(30) + (370)/(30) = (478)/(30)`
1/2(b1 + b2) =

`(478)/(60)`
We now have a =

`(478)/(60) √(5)`
Assume a is rational.

Then for some integers x and y, y not equal to zero


`(x)/(y) = (478)/(60) √(5)`
Dividing both sides by

`(478)/(60)`
we obtain

`(60x)/(478y) = √(5)`
Since a rational number cannot equal an irrational number our original assumption that a was rational was false.

a is irrational.