(c.) Determine the surface area of the box in terms of \( x \). (b) Determine the volume of the box in terms of \( x \). (c) Determine the surface area in terms of \( x \) if the breadth a is doubled. Determine the volume in terms of \( x \) if the length is one (te original length.

To solve the problem, we need to define the dimensions of the box in terms of \( x \). Let's assume the following: - Length of the box: \( l = x \) - Breadth of the box: \( b \) - Height of the box: \( h \) ### (a) Surface Area of the Box in Terms of \( x \) The surface area \( S \) of a rectangular box is given by the formula: \[ S = 2(lw + lh + wh) \] Substituting \( l = x \): \[ S = 2(xb + xh + bh) \] ### (b) Volume of the Box in Terms of \( x \) The volume \( V \) of a rectangular box is given by the formula: \[ V = l \cdot b \cdot h \] Substituting \( l = x \): \[ V = x \cdot b \cdot h \] ### (c) Surface Area in Terms of \( x \) if the Breadth is Doubled If the breadth \( b \) is doubled, the new breadth becomes \( 2b \). The new surface area \( S' \) is: \[ S' = 2(x(2b) + xh + (2b)h) \] Simplifying this: \[ S' = 2(2xb + xh + 2bh) = 4xb + 2xh + 4bh \] ### (d) Volume in Terms of \( x \) if the Length is One (the Original Length) If the length is set to 1, then we have: \[ l = 1 \] The volume \( V' \) becomes: \[ V' = 1 \cdot b \cdot h = b \cdot h \] ### Summary - **Surface Area in terms of \( x \)**: \( S = 2(xb + xh + bh) \) - **Volume in terms of \( x \)**: \( V = x \cdot b \cdot h \) - **Surface Area if breadth is doubled**: \( S' = 4xb + 2xh + 4bh \) - **Volume if length is one**: \( V' = b \cdot h \)