a Recall the formulas for the perimeter and the area of a rectangle.
B
b Substitute 12 for P into the perimeter formula and simplify.
C
c Use the table from the previous part and plot the points.
D
d The graph we made in the previous part can be helpful.
E
e You can use the table to find the length and width that maximize and minimize the area.
A
a
P=2(x+y) A=xy
B
b
x
y
A
1
5
5
2
4
8
3
3
9
4
2
8
5
1
5
C
c
D
d As the length changes, the area of a rectangle behaves like the quadratic function.
E
e See solution.
Practice makes perfect
a We are asked to write equations for the perimeter and area of a rectangle that has a length of x and a width of y. Let's recall that the perimeter of a rectangle is two times the sum of its length and width.
P=2( x+ y)
Next, let's recall that the area of a rectangle is the product of its length and width.
A= x y
b In this part we are asked to tabulate all possible whole-number values for the length and width of the rectangle and find the area for each pair. Since we are given that a rectangle has a perimeter of 12 units, let's substitute this value for P into the perimeter formula.
This means that we need to write down pairs of numbers that added gives 6. Then we will multiply them to evaluate the area for each pair.
x
y
A=xy
1
5
1* 5=5
2
4
2* 4=8
3
3
3* 3=9
4
2
4* 2=8
5
1
5* 1=5
c Now we want to graph the area of the rectangle with respect to its length. To do this we will use the table we made in the previous part and plot the points on a coordinate plane. The length will be on the horizontal axis and the area will be on the vertical axis.
Notice that the length must be greater than 0 and less than 6.
d Looking at the graph we made in the previous part, we can see that the graph seems to present a quadratic function. Therefore we can assume that as the length of a rectangle changes the area behaves like a quadratic function.
e If we look at the table or a graph we can see that the area is the greatest when the length and the width are equal. Also, we can see the area is the least when the difference between the length and the width is the greatest.
