The area of a rectangle is 45 cm². two squares are constructed such that two adjacent sides of the rectangle are each also the side of one of the squares. the combined area of the two squares is 106 cm². find the lengths of the sides of the squares.

5 cm, 9 cm

Step-by-step explanation:

If x and y are the lengths of sides of the rectangle, then the problem statement tells you ...

Substituting y = 45/x into the second equation, you get

... x² + (45/x)² = 106

Multiplying by x² gives ...

... x⁴ -106x² +2025 = 0 . . . . . after subtracting 106x²

... (x² -53)² -784 = 0 . . . . . . . . complete the square

... x² = 53 ±√784 = 53 ±28 . . . solve for x²

... x = √81 or √25 = 9 or 5 . . . . solve for x (which must be positive)

y is the other of these two values, so ...

The lengths of the sides of the squares are 5 cm and 9 cm.

_____

Comment on the graph

The graph is symmetrical about the origin, so there are also two solutions at (x, y) = (-5, -9) and (-9, -5). Since x and y represent distances, these are extraneous solutions with respect to the original question.

  b.   the average rate of change of f(x) is less than the average rate of change of g(x).

step-by-step explanation:

since the length of the interval is the same in each case, we only need to compare the change in y-value to see which is greater.

so the change in y-value for function f is (6 )) = 9.

the table tells us

so the change in y-value for function g is (31 -1) = 30.

the change in y-value of f(x) on the interval is less than the change in y-value of g(x) on the same interval, so the average rate of change of f(x) is less than the average rate of change of g(x).