Question

which of the solutions to the following problems represent a rational number? select all that apply. a rectangle abcd has a length of 5 cm, and a width of 2√8 cm. what is the perimeter of the rectangle? b the area of a rectangular swimming pool is 37 yards²; its width is 3 yards. what is the length of the swimming pool? c alan divides a rope with a length of 2√2 m into 3 equal pieces. what is the length of each piece? d a square garden has a length of √2 ft. what is the area of the garden?

Explanation:

Step1: Analyze Option A

First simplify width: $2\sqrt{8}=2\times2\sqrt{4}=4\sqrt{2}$
Perimeter formula: $P=2(l+w)=2(5+4\sqrt{2})=10+8\sqrt{2}$
This includes irrational $\sqrt{2}$, so not rational.

Step2: Analyze Option B

Length formula: $l=\frac{A}{w}$
Substitute values: $l=\frac{37}{3}$
This is a ratio of integers, so rational.

Step3: Analyze Option C

Piece length formula: $l=\frac{\text{Total length}}{3}$
Substitute values: $l=\frac{2\sqrt{2}}{3}$
This includes irrational $\sqrt{2}$, so not rational.

Step4: Analyze Option D

Area of square: $A=s^2$
Substitute $s=\sqrt{2}$: $A=(\sqrt{2})^2=2$
2 is an integer, so rational.

Answer:

B. The area of a rectangular swimming pool is 37 yards²; its width is 3 yards. What is the length of the swimming pool?
D. A square garden has a length of $\sqrt{2}$ ft. What is the area of the garden?