We are told that the stage manager of a school play creates a rectangular stage that has whole number dimensions and an area of 42 square yards. We want to find the least number of yards of string lights needed to enclose the stage.
To do so, let's recall the formula for the area of a rectangle.
A= l wHere, l indicates the length and w is the width of the rectangle. Now, remember that a factor pair of a number is two integers that result in the given number multiplied together.
Notice that the factor pairs have the same form as the formula for the area of a rectangle. This means that we can interpret the factor pairs as the product of the length and the width of the rectangle. Since the area of the rectangular stage is 42 square yards, the factor pairs of this number will be the possible side lengths of the rectangular stage. Let's list them.
42 &= 1 * 42
42 &= 2 * 21
42 &= 3 * 14
42 &= 6 * 7
This means that we will have 4 possible arrangements for the rectangular stage.
1 yard long and 42 yards wide
2 yards long and 21 yards wide
3 yards long and 14 yards wide
6 yards long and 7 yards wide
Note that the lengths and widths could be reversed, but the yardage of string lights will be the same. Therefore, we will only consider each combination once. To find the number of yards of string lights that we need for enclose the stage, we can calculate the perimeter of the rectangular stage. Let's recall the formula for the perimeter of a rectangle.
P = 2 l + 2 w
We will substitute the obtained possible arrangements in the above formula to find the possible perimeters of the stage.
l, w
P=2l + 2 w
P
l= 1, w= 42
P=2( 1) + 2( 42)
86
l= 2, w= 21
P=2( 2) + 2( 21)
46
l= 3, w= 14
P=2( 3) + 2( 14)
34
l= 6, w= 7
P=2( 6) + 2( 7)
26
From the table, we can see that the smallest value of the perimeter is 26 yards. Therefore, we need at least 26 yards of string lights.
