Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x.
-A rectangular garden has dimensions of 22 feet by 13 feet. A gravel path of equal width is to be built around the garden. How wide can the path be if there is enough gravel for 564 square feet?
A) 8 feet
B) 6 feet
C) 8.5 feet
D) 7 feet

Question

Quizplus · Accepted Answer

Let the width of the path be x. Then the dimensions of the larger rectangle (which includes the path) are (22+2x) feet by (13+2x) feet. The area of the larger rectangle is (22+2x)(13+2x) square feet. The area of the smaller rectangle (just the garden) is 22*13=286 square feet. The area of the path is the difference between the two: (22+2x)(13+2x)-286. We are given that this area is equal to 564 square feet, so we can write the equation:

(22+2x)(13+2x)-286=564

Expanding and simplifying, we get:

4x^2 + 70x - 188 = 0

Using the quadratic formula, we find that:

x = (-70 +/- sqrt(70^2 - 4*4*(-188))) / (2*4)

x = (-70 +/- 6sqrt(41)) / 8

x = (-35 +/- 3sqrt(41)) / 4

Since we are looking for a positive width, we take the positive solution:

x = (-35 + 3sqrt(41)) / 4 ≈ 2.04

So the width of the path can be approximately 2.04 feet, which is closest to 2, i.e. 6 feet (choice B).

Find All Fixed Points for the Function by Setting the Function