Final answer:
The problem is an optimization question in mathematics, where we aim to minimize the area of a rectangular flower garden and its surrounding brick borders. After defining the dimensions of the garden as variables, we construct an equation for total area including the borders. We would then use calculus to find the derivative, set it to zero to find the minimum area.
Step-by-step explanation:
The question revolves around a classic optimization problem in mathematics, particularly within the study of calculus. We want to minimize the combined area of a rectangular flower garden and its borders, where the garden has an area of 6 square feet, and is surrounded by a brick border that is 1 foot wide on two sides and 6 feet wide on the other two sides.
To solve this problem, we first need to define variables for the dimensions of the garden. Let's use x and y for the width and length of the garden, respectively. From the problem statement, we know the area of the garden, which gives us the equation xy = 6. The total area including the borders would be expressed as (x + 2)(y + 14). The 2 comes from the 1-foot wide borders on two sides and the 14 comes from adding two times the 6-foot and 1-foot borders.
Next, since we have xy = 6, we can solve for one variable in terms of the other, say y = 6/x. We can then substitute this into the total area equation to express the area solely in terms of x: A(x) = (x + 2)((6/x) + 14). To find the minimum area, we'd need to take the derivative of A(x) with respect to x, set it to zero, and solve for x. After finding the optimal value of x, we substitute it back into y = 6/x to find the corresponding value of y that minimizes the total area.
To fully answer the student's question, a calculus-based solution involving finding the derivative of A(x), setting it to zero, and solving would be necessary. However, this could be an extensive process that would include testing the critical points to ensure that we find the minimum rather than maximum and involve second derivative tests or a closed interval method.