Betty decides to compare the diagonals of both the square and rectangular tables customers will use in the coffee shop. Her measurements show that the square tables have a side length of 34 inches and the rectangular tables are 29 1/3inches wide and 35 inches long. She uses the formula d i a g o n a l space equals space square root of left parenthesis w i d t h right parenthesis squared plus left parenthesis h e i g h t right parenthesis squared end root to compute the diagonal of each table.
Which table has a diagonal whose length (measured in inches) is an irrational number? What is the length of this diagonal?
The diagonal of the square is √(34^2 + 34^2)
= √2312 inches
so what is your verdict?
repeat for the other one. Suggestion, do not use decimals
Betty decides to compare the diagonals of both the square and rectangular tables customers will use in the coffee shop. Her measurements show that the square tables have a side length of 34 inches and the rectangular tables are inches wide and 35 inches long. She uses the formula to compute the diagonal of each table.
To determine which table has a diagonal whose length is an irrational number, we need to apply the formula for calculating the diagonal and then check if the resulting value is irrational.
For the square table:
Side length = 34 inches
The formula for the diagonal of a square is given by:
Diagonal = √(Side length^2 + Side length^2)
Substituting the value:
Diagonal = √(34^2 + 34^2)
Diagonal = √(2,896 + 2,896)
Diagonal = √5,792
Diagonal = 76 inches
For the rectangular table:
Width = 29 1/3 inches
Length = 35 inches
The formula for the diagonal of a rectangle is given by:
Diagonal = √(Width^2 + Length^2)
Substituting the values:
Diagonal = √((29 1/3)^2 + 35^2)
Diagonal = √((29.33)^2 + 35^2)
Diagonal = √(861.0889 + 1225)
Diagonal = √2086.0889
Diagonal ≈ 45.68 inches
Now, let's determine which diagonal has a length that is an irrational number.
The diagonal length of the square table is 76 inches, which is a rational number because it can be expressed as a fraction.
The diagonal length of the rectangular table is approximately 45.68 inches, which is an irrational number. It cannot be expressed as a fraction and has infinite non-repeating decimal places.
Therefore, the rectangular table has a diagonal whose length is an irrational number, and the length of this diagonal is approximately 45.68 inches.