Final answer:
To find the dimensions that yield the minimum surface area for the tank, we need to consider the volume and surface area formulas for a rectangular tank.
By using calculus, we can differentiate the surface area equation to find the values of length, width, and height that minimize the surface area.
Step-by-step explanation:
To find the dimensions that yield the minimum surface area for the tank, we need to consider the volume and surface area formulas for a rectangular tank. Let's assume the length, width, and height of the tank are l, w, and h, respectively. The volume of the tank is given as 61 ft³, so we have lwh = 61.
The surface area of the tank can be calculated using the formula SA = 2lw + 2lh + 2wh. To find the minimum surface area, we optimize this function by using calculus.
By differentiating SA with respect to l, w, and h, and setting the resulting equations equal to zero, we can solve for the values of l, w, and h that minimize the surface area. The solutions will give us the dimensions that yield the minimum surface area for the tank.