Question

Will the values described in each situation be rational or irrational?



Select Rational or Irrational to describe each situation.

Rational Irrational

R or D the length of a rectangle with a rational area and irrational width
R or D the area of a circle with a rational radius
R or D the perimeter of a square with irrational side lengths
R or D the volume of a cube with rational side lengths

Answer 1

Help

Answer 2

rational numbers are numbers that can be written as fractions, using integers
irrational numbers are numbers that can not

if we have an irrational width, but rational area, then we would have to find some number when multiplied by an irrational number, gives a rational number
area=length x width

let x be the irrational number, and a/b, c/d be rational numbers, a,b,c,d, exist in the set of integers, where c/d is the area which is the rational
xa/b=c/d,
multiply by b/a
x=cb/ad

so this shows an irrational width, multiplied by a rational length, to give a rational area, but if this were true, then we have the irrational number x=cb/ad, since a/b, c/d are rational, meaning htey can be written as a fraction of integers
meaning x wouldn't be irrational, since we have shown it in a form of fraction with integers.

Hence, our length must be irrational also




Area of a circle = pir^2

we have a rational radius, meaning r=a/b, a,b exist in set of integers
A=pi(a/b)^2

so we have an irrational, multiplied by a rational
using part 1 before, we showed that a rational x irrational does not give a rational answer, so A, area, is irrational




perimeter of a square, irrational side lengths
perimeter is addition of all side lengths

let side length = x, and perimeter=4x

we know 4 is a rational number, since 4 can be written as a fraction, using integers, 4/1, hence 4 is rational
x is the irrational side length
rational x irrational is irrational
hence perimeter is irrational




volume of a cube = a^3
we know that a is rational, hence a^3 = (a/b)^3, a^3/b^3
a,b must be integers to be rational, hence a^3/b^3 is rational, hence volume is rational

Answer 3

just to fix the picture I added