Answer
a) $12x + 4$
b) No. $h>2\pi$
Work Step by Step
a) We are given:
- square: side length $=6x+2$
- rectangle: width $=3x+1$
- the areas are equal
The area of the square is:
$A_{square}=(6x+2)^2$
The area of the rectangle is:
$A_{rectangle}=(3x+1)L,$
where $L$ is the length of the rectangle (what we need to find).
As the two areas are equal, we have:
$(3x+1)L=(6x+2)^2$
$L=\frac{2^2(3x+1)^2}{(3x+1)}=4(3x+1)=12x+4$
b) The height of the rectangle prism is
$h=\frac{\pi(a+4)}{2}$
If $h=1$, we calculate $a$:
$2(1)=\pi(a+4)$
$a+4=\frac{2}{\pi}$
$a=\frac{2}{\pi}-4\approx -3.36<-0$
Therefore $h$ cannot be $1$ feet.
We must have:
$\frac{2h}{\pi}-4>0$
$2h>4\pi$
$h>2\pi$
