Sure, here’s a fun math problem for you:
Problem:
You have a rectangular garden that measures 20 meters in length and 15 meters in width. You want to create a walkway of uniform width around the garden. The area of the walkway should be equal to the area of the garden.
Question:
What should be the width of the walkway?
Solution:
- Let the width of the walkway be ( x ) meters.
- The dimensions of the garden with the walkway will be ( (20 + 2x) ) meters by ( (15 + 2x) ) meters.
- The area of the garden is ( 20 \times 15 = 300 ) square meters.
- The area of the garden with the walkway is ( (20 + 2x) \times (15 + 2x) ) square meters.
- The area of the walkway alone is the area of the garden with the walkway minus the area of the garden:
[ (20 + 2x)(15 + 2x) - 300 ]
- We want the area of the walkway to be equal to the area of the garden, so:
[ (20 + 2x)(15 + 2x) - 300 = 300 ]
- Simplify and solve for ( x ):
[ (20 + 2x)(15 + 2x) = 600 ] [ 300 + 30x + 40x + 4x^2 = 600 ] [ 4x^2 + 70x + 300 = 600 ] [ 4x^2 + 70x - 300 = 0 ]
- Solve the quadratic equation:
[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ] [ a = 4, ; b = 70, ; c = -300 ] [ x = \frac{-70 \pm \sqrt{70^2 - 4 \cdot 4 \cdot (-300)}}{2 \cdot 4} ] [ x = \frac{-70 \pm \sqrt{4900 + 4800}}{8} ] [ x = \frac{-70 \pm \sqrt{9700}}{8} ] [ x = \frac{-70 \pm 98.49}{8} ]
- The two possible solutions are:
[ x = \frac{28.49}{8} = 3.56 ] [ x = \frac{-168.49}{8} = -21.06 ] (This is not a valid solution because the width cannot be negative)
So, the width of the walkway should be approximately 3.56 meters.