a.
[tex]h = 16x + 12y[/tex]
[tex]l = 7x + 29y[/tex]
[tex]w = 12x - 5y[/tex]
to find the volume of a cuboid, we use
[tex]v = h \times l \times w[/tex]
[tex]v = (16x + 12y)(7x + 29y)(12 - 5y)[/tex]
[tex] = (112 {x}^{2} + 464xy + 84xy + 348 {y}^{2} )(12x - 5y)[/tex]
[tex] = (112 {x}^{2} + 548xy + 348 {y}^{2} )(12x - 5y)[/tex]
[tex] = 1344 {x}^{3} - 560 {x}^{2} y + 6576 {x}^{2} y - 2740x {y}^{2} + 4176x {y}^{2} - 1740 {y}^{3} [/tex]
[tex] = 1344 {x}^{3} + 6016 {x}^{2} y + 1436x {y}^{2} + 1740 {y}^{3} [/tex]
b.
[tex]w = 13a - 42b + 15[/tex]
[tex]l = 42a + 2b - 31[/tex]
to find the area of a rectangle, we use
[tex]a = wl[/tex]
[tex]a = (13a - 42b + 15)(42a + 2b - 31)[/tex]
[tex] = 546 {a}^{2} + 26ab - 403a - 1764ab - 84 {b}^{2} + 1302b + 630a + 30b - 465[/tex]
[tex] = 546 {a}^{2} + 227a - 1738ab + 1332b - 84 {b}^{2} - 465[/tex]
c.
[tex]s = 17d - 2[/tex]
to find the volume of a cube, we use
[tex]v = {s}^{3} [/tex]
[tex]v = {(17d - 2)}^{3} [/tex]
[tex] = (17d - 2)(17d - 2)(17d - 2)[/tex]
[tex] = (17d - 2)(289 {d}^{2} - 34d - 34d + 4)[/tex]
[tex] = (17d - 2)(289 {d}^{2} - 68d + 4)[/tex]
[tex] = 4913 {d}^{3} - 578 {d}^{2} + 68d - 578 {d}^{2} + 136d - 8[/tex]
[tex] = 4913 {d}^{3} - 1156 {d}^{2} + 204d - 8[/tex]
