Question

Find a formula for the described function. an open rectangular box with volume 9 m3 has a square base. express the surface area sa of the box as a function of the length of a side of the base, x. sa = x2+ 36 x​ m2 state the domain of sa.

Answer 1

Let

x--------> the length side of the square base

h--------> the height of the box

we know that

the volume of the box is equal to

$V=x^{2} *h\\ V=9\ m^{3}$

so

$x^{2} *h=9\\\\h=\frac{9}{x^{2} }$

the surface area of the box is equal to

$SA=area\ of\ the\ base+perimeter\ of\ base*height$ (remember that the box is open)

area of the base=$(x^{2})\ m^{2}$

Perimeter of the base=$(4*x)\ m$

height=(h) m

$h=\frac{9}{x^{2}}$

substitute

$SA=x^{2} +4*x*h\\ \\ \\ SA=x^{2} +4*x*\frac{9}{x^{2} } \\ \\ SA=x^{2} +\frac{36}{x}$

we know that

the value of x can not be negative and the denominator can not be zero

therefore

the answer is

the domain of  SA is x> 0

the domain is the interval-------------> (0,∞)