Final answer:
By establishing the width of the sandbox as 'w' and accounting for the strip of grass, we use the quadratic equation to find that the dimensions of the sandbox are 3 feet in width and 5 feet in length.
Step-by-step explanation:
To find the dimensions of the sandbox, we first need to establish a variable for the width of the sandbox. Let's call this width 'w' feet. According to the problem, the length of the sandbox is 'w + 2' feet. Since the strip of grass is 5 feet wide on all sides, the total width of the grass and sandbox together is 'w + 2(5)' and the total length is 'w + 2 + 2(5)'. This results in a total width of 'w + 10' feet and a total length of 'w + 12' feet.
The area of the grass and sandbox together is given as 189 square feet. Thus, we have the equation
(w + 10)(w + 12) = 189. Expanding this equation, we get w^2 + 22w + 120 = 189. Simplifying, we subtract 189 from both sides, resulting in w^2 + 22w - 69 = 0. Factoring this quadratic equation, we find two potential solutions for w, but only one will make sense in the context of the problem (because the dimensions can't be negative).
Let's assume 'w' equals 3 feet, which is the sensible solution. Thus, the dimensions of the sandbox are: width is 3 feet, and length is 'w + 2', which is 5 feet.