Answer:
The surface area of the gift box is = [tex]A = \frac{1188}{7}x^{2}+ \frac{774}{7}x +\frac{132}{7}[/tex]
Step-by-step explanation:
The surface area of the cylinder can be calculated using this formula:
[tex]A=2\pi rh+2\pi r^2[/tex]
in this case, r is not a number, but rather an expression, which is r = [tex]3x +1[/tex].
The height of the gift box is twice the radius, which is h = [tex]2(3x+1)= 6x+2[/tex]
To get our curved surface area, we carefully put the expressions for h and r into the equation.
[tex]A = 2 \times \frac{22}{7} \times (3x+1) \times (6x+2) + 2 \times \frac{22}{7} \times (3x+1)^{2}[/tex]
[tex]A = \frac{44}{7} (3x+1) \times (6x +2) + \frac{44}{7} (9x^{2} + 6x +1)\\A =\frac{132}{7}x + \frac{44}{7} \times (6x+2)+ \frac{396}{7}x^{2}+\frac{246}{7}x +\frac{44}{7}[/tex]
[tex]A = \frac{792}{7}x^{2} +\frac{264}{7}x +\frac{264}{7}x + \frac{88}{7} + \frac{396}{7}x^{2} + \frac{246}{7}x + \frac{44}{7}[/tex]
[tex]A = \frac{1188}{7}x^{2}+ \frac{774}{7}x +\frac{132}{7}[/tex]
The surface area of the gift box is = [tex]A = \frac{1188}{7}x^{2}+ \frac{774}{7}x +\frac{132}{7}[/tex]
