a rectangle has an area of 48 square yards. For what whole number values of length and width will the rectang,e have the smallest perimeter?
What do you think?
What two whole numbers will multiply to give you 48?
To find the whole number values of length and width for which the rectangle has the smallest perimeter, we need to consider the relationship between the area and the perimeter of a rectangle.
The area of a rectangle is given by the formula: Area = Length × Width
The perimeter of a rectangle is given by the formula: Perimeter = 2 × (Length + Width)
Given that the area is 48 square yards, we can start by finding the factors of 48 that are whole numbers. The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
We now need to find pairs of factors that satisfy the condition for the smallest perimeter. Since the perimeter is given by the formula: Perimeter = 2 × (Length + Width), we want to find a pair of factors that minimizes the sum of Length and Width.
Let's try out different pairs and calculate the sum of Length and Width for each:
Pair (Length, Width) | Sum (Length + Width)
------------------------------------------------
(1, 48) | 49
(2, 24) | 26
(3, 16) | 19
(4, 12) | 16
(6, 8) | 14
(8, 6) | 14
...
...
From the calculations, we can see that the pair (4, 12) yields the smallest sum of Length and Width, which is 16. Therefore, the whole number values of length and width for which the rectangle has the smallest perimeter are 4 and 12.