Question

show that each conjecture is false by providing a counterexample. part 1 of 2 (a) if the area of a rectangle is 16, then the length is 4 and the width is 4. counterexample: length =, width = part 2 of 2 (b) if the perimeter of a rectangle is 48, then the length is 12 and the width is 12. counterexample: length =, width =

Explanation:

Step1: Recall area formula for rectangle

The area formula of a rectangle is $A = l\times w$, where $A$ is the area, $l$ is the length and $w$ is the width.

Step2: Find counter - example for area conjecture

We need to find non - 4 and non - 4 values for length and width such that $l\times w=16$. Let $l = 8$ and $w = 2$ since $8\times2=16$.

Step3: Recall perimeter formula for rectangle

The perimeter formula of a rectangle is $P=2(l + w)$.

Step4: Find counter - example for perimeter conjecture

We know that $P = 48$, and $P=2(l + w)$, so $l + w=\frac{P}{2}=24$. We can choose $l = 20$ and $w = 4$ since $2(20 + 4)=2\times24 = 48$ and $l
eq12$ and $w
eq12$.

Answer:

(a) length = 8, width = 2
(b) length = 20, width = 4