Final answer:
Luke should increase each dimension of his deck by solving the quadratic equation derived from setting the new area as three times the original area, considering the new dimensions as (22 + x) feet by (11 + x) feet.
Step-by-step explanation:
To calculate by how much Luke should increase each dimension of his deck to triple its area, we begin with the dimensions of the original deck, which are 22 feet by 11 feet. The original area is therefore 22 feet x 11 feet = 242 square feet. To triple the area, the new area must be 3 x 242 square feet = 726 square feet.
Let's assume Luke increases each dimension by a length of x feet. The new dimensions will be (22 + x) feet by (11 + x) feet. The equation for the new area then becomes (22 + x)(11 + x) = 726. Expanding this, we get 242 + 33x + x^2 = 726. Simplifying, we find x^2 + 33x - 484 = 0, which is a quadratic equation.
Upon solving this quadratic equation, we find that x has two potential values. However, we discard the negative solution as it doesn't make sense in the context of the problem. Hence, the positive value of x gives us the amount by which Luke should increase each dimension of his deck.