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/ 3 inches wide and 35 inches long. She uses the formula d i a g o n a l (width Square root)^2 + (h e i g h t)^2 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?
sqrt (34^2 + 34^2)
= sqrt [(2*17)^2 + (2*17)^2]
= sqrt [ 4*17^2 + 4*17^2)]
=2 * sqrt (2*17^2)
= 34 sqrt 2 which is irrational
sqrt [(29 1/3)^2 + 35^2 ]
sqrt [ (88/3)^2 + 35^2 ]
sqrt [ 7744/9 + 1225 ]
sqrt [ 7744/9 + 11025/9 ]
sqrt [ 18769/9 ]
(1/3) sqrt (18769)
(1/3)(137)
= 137/3 which is rational
To determine which table has a diagonal with an irrational length and calculate the length, we can use the given formula for the diagonal:
diagonal = √(width^2 + height^2)
1. For the square table:
The side length of the square table is given as 34 inches. Since all sides of a square are equal, the width and height of the square table are both 34 inches. We can substitute these values into the formula:
diagonal = √(34^2 + 34^2)
diagonal = √(1156 + 1156)
diagonal = √(2312)
diagonal ≈ 48.08 inches
2. For the rectangular table:
The width of the rectangular table is given as 29 1/3 inches and the length is 35 inches. We can convert the mixed fraction to a decimal:
29 1/3 = 29 + 1/3 = 29 + 0.333...
Thus, the width is approximately 29.333... inches. We can substitute these values into the formula:
diagonal = √((29 1/3)^2 + 35^2)
diagonal = √((29.333...)^2 + 35^2)
diagonal = √(861.110... + 1225)
diagonal = √(2086.110...)
Since the width and length of the rectangular table are not perfect square numbers, the diagonal is an irrational number.
The rectangular table has a diagonal with an irrational length. However, we cannot determine the exact length unless we know the precise value of √(2086.110...).