Final answer:
The width of the den is 11 feet and the length is 19 feet.
Step-by-step explanation:
Let x be the width of the den. According to the problem, the length is 8 feet longer than the width, so the length would be x + 8. The area of a rectangle is given by multiplying the length by the width, so we can set up the equation:
x(x + 8) = 209
Expanding the equation gives us a quadratic equation:
x^2 + 8x - 209 = 0
We can solve this equation by factoring or using the quadratic formula. By factoring, we find:
(x - 11)(x + 19) = 0
Setting each factor equal to zero, we find two possible values for x: x = 11 and x = -19. Since the dimensions of the den can't be negative, the width of the den is 11 feet. Therefore, the length of the den would be 11 + 8 = 19 feet.