30 write and simplify a polynomial expression for the surface area of each figure.

S.A= 16x² + 30x + 6

S.A = 4πw² - 4π

S.A = 19y² + 14y

Step-by-step explanation:

The figure is rectangular prism

∵ S.A = perimeter of base × height + 2 base area

∵ Perimeter of the base = 2(x + 3 + 2x + 1) = 2(3x + 4)

                                        = 6x + 8

∵ Area of the base = (x + 3)(2x + 1) = 2x² + x + 6x + 3

                               = 2x² + 7x + 3

∴ S.A = (6x + 8)(2x) + 2(2x² + 7x + 3)

         = 12x² + 16x + 4x² + 14x + 6

         = 16x² + 30x + 6

The figure is cylinder

∵ S.A = perimeter of base × height + 2 base area

∵ Perimeter of the base = 2π(w - 1) = 2πw - 2π

∵ Area of the base = π(w - 1)² = π(w² - 2w + 1)

                               = πw² - 2πw + π

∴ S.A = (2πw - 2π)(w + 3) + 2(πw² - 2πw + π)

         =2πw² + 6πw - 2πw - 6π + 2πw² - 4πw + 2π

         = 4πw² - 4π

The figure is triangular prism

∵ S.A = perimeter of base × height + 2 base area

∵ Perimeter of the base = 2y + 2y + 1 + y + 3 = 5y + 4

∵ Area of the base = 1/2(2y)(2y + 1) =y(2y + 1)

                                = 2y² + y

∴ S.A = (5y + 4)(3y) + 2(2y² + y)

         = 15y² + 12y + 4y² + 2y

         = 19y² + 14y

set each factor equal to 0

4x-3 = 0

2x-1 = 0

solve for x.

4x-3=0

4x=3

x= 3/4

solve for x. 

2x-1= 0

2x= 1

x= 1/2

your answer should be 1/2 is less than or equal to x and 3/4 is greater than or equal to x