To determine the length [tex]\( c \)[/tex] of the diagonal cut made on a 4 ft by 8 ft piece of plywood, we'll use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the diagonal cut in our case) is equal to the sum of the squares of the lengths of the two other sides.
Let's denote:
- [tex]\( a \)[/tex] as the length of one side of the rectangle,
- [tex]\( b \)[/tex] as the length of the other side of the rectangle,
- [tex]\( c \)[/tex] as the length of the diagonal cut.
In this scenario, we have:
- [tex]\( a = 4 \)[/tex] ft,
- [tex]\( b = 8 \)[/tex] ft.
We can substitute the given values into the Pythagorean theorem formula:
[tex]\[ c^2 = a^2 + b^2 \][/tex]
First, calculate the squares of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ a^2 = 4^2 = 16 \][/tex]
[tex]\[ b^2 = 8^2 = 64 \][/tex]
Next, sum these squared values to get [tex]\( c^2 \)[/tex]:
[tex]\[ c^2 = 16 + 64 = 80 \][/tex]
To find [tex]\( c \)[/tex], we need to take the square root of [tex]\( 80 \)[/tex]:
[tex]\[ c = \sqrt{80} \][/tex]
Thus, the length [tex]\( c \)[/tex] of the diagonal cut is [tex]\( \sqrt{80} \)[/tex] feet.
Given the multiple-choice options, the correct answer is:
[tex]\[ \sqrt{80} \, \text{feet} \][/tex]
