(b) Determine the volume of the box in terms of \( x \). (c) Determine the surface area in terms of \( x \) if the breadth and height is doubled. (d) Determine the volume in terms of \( x \) if the length is one third of the original length. If the length is multiplied by a scale factor of \( k \), by what number in terms of \( k \) will the volume of the above prism be multiplied by? If the length and breadth is multiplied by a scale factor of \( k \), by what number in terms of \( k \) will the volume of the above prism be multiplied by? (f) If the length, breadth and height is multiplied by a scale factor of \( k \), by what number in terms of \( k \) will the volume of the above (h) If the length, breadth and height is multiplied by a scale factor of (h) by what number in terms of \( k \) will the surface area of the above prism be multiplied by?

To address your questions, let's denote the dimensions of the box (prism) as follows: - Length = \( l \) - Breadth = \( b \) - Height = \( h \) ### (b) Determine the volume of the box in terms of \( x \). The volume \( V \) of a box is given by the formula: \[ V = l \times b \times h \] If we express the dimensions in terms of \( x \), we can write: \[ V = f(x) \quad \text{(where \( f(x) \) is a function of \( x \))} \] ### (c) Determine the surface area in terms of \( x \) if the breadth and height are doubled. The surface area \( S \) of a box is given by the formula: \[ S = 2(lb + bh + hl) \] If the breadth and height are doubled, the new dimensions become: - New breadth = \( 2b \) - New height = \( 2h \) The new surface area \( S' \) becomes: \[ S' = 2(l(2b) + (2b)(2h) + (2h)l) = 2(2lb + 4bh + 2hl) = 4(lb + 2bh + hl) \] Thus, the surface area is multiplied by 2. ### (d) Determine the volume in terms of \( x \) if the length is one third of the original length. If the length is reduced to one third, the new length becomes: - New length = \( \frac{1}{3}l \) The new volume \( V' \) is: \[ V' = \left(\frac{1}{3}l\right) \times b \times h = \frac{1}{3}V \] Thus, the volume is multiplied by \( \frac{1}{3} \). ### If the length is multiplied by a scale factor of \( k \), by what number in terms of \( k \) will the volume of the above prism be multiplied by? If the length is multiplied by \( k \), the new volume \( V' \) becomes: \[ V' = (kl) \times b \times h = k \times V \] Thus, the volume is multiplied by \( k \). ### If the length and breadth are multiplied by a scale factor of \( k \), by what number in terms of \( k \) will the volume of the above prism be multiplied by? If both the length and breadth are multiplied by \( k \), the new volume \( V' \) becomes: \[ V' = (kl) \times (kb) \times h = k^2 \times V \] Thus, the volume is multiplied by \( k^2 \). ### If the length, breadth, and height are multiplied by a scale factor of \( k \), by what number in terms of \( k \) will the volume of the above prism be multiplied by? If all three dimensions are multiplied by \( k \), the new volume \( V' \) becomes: \[ V' = (kl) \times (kb) \times (kh) = k^3 \times V \] Thus, the volume is multiplied by \( k^3 \). ### (h) If the length, breadth, and height are multiplied by a scale factor of \( k \), by what number in terms of \( k \) will the surface area of the above prism be multiplied by? If all three dimensions are multiplied by \( k \), the new surface area \( S' \) becomes: \[ S' = 2((kl)(kb) + (kb)(kh) + (kh)(kl)) = 2(k^2 lb + k^2 bh + k^2 hl) = k^2 S \] Thus, the surface area is multiplied by \( k^2 \). In summary: - Volume with length one third: \( \frac{1}{3}V \) - Volume with length multiplied by \( k \): \( kV \) - Volume with length and breadth multiplied by \( k \): \( k^2V \) - Volume with length, breadth, and height multiplied by \( k \): \( k^3V \) - Surface area with length, breadth, and height multiplied by \( k \): \( k^2S \)