Final answer:
To find the maximum capacity of a rectangular prism box, solve the surface area equation in terms of one variable and use it to find the maximum volume.
Step-by-step explanation:
To find the maximum capacity of the rectangular prism box, we need to know the dimensions of the box. Let's assume the length, width, and height of the box are l, w, and h respectively. The total surface area of a rectangular prism is given by the formula: 2lw + 2lh + 2wh = 294 cm². Since we want to maximize the capacity, we can set the surface area formula equal to the volume formula: lwh = V. Rearranging the surface area equation, we have lw + lh + wh = 147 cm². Now, we can use this equation to find the maximum volume by maximizing two of the variables while keeping the third variable constant. For example, if we want to maximize l and w, we can set h = 1, and solve for l and w as follows:
- lw + lh + wh = 147 (Substitute h = 1)
- lw + l + w = 147 (Simplify)
- l(w + 1) + w = 147 (Factor out l)
- l(w + 1) = 147 - w (Rearrange)
- l = (147 - w)/(w + 1) (Divide by w + 1)
Now, you can substitute different values for w to find the corresponding values of l and calculate the corresponding volumes. Repeat the process for other combinations of variables to find the maximum volume. Remember, the units for the volume will be cm³.
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