1.Consider divided by . (a)Write a real-world problem for the division. (b)Create a model or write an equation for the division. Find the quotient for the real-world problem in part (a). Show your work or explain your reasoning

X = side length of square base
h = height

area of base = x^2
volume = height*(area of base)
volume = h*x^2
245 = h*x^2
h = 245/(x^2)



Surface area

SA = x^2+4*h*x
SA = x^2+4*(245/(x^2))*x
SA = x^2+4*(245/x)
SA = x^2+4*(245/x)
S(x) = x^2+(980/x)

Goal is to minimize S(x)



S(x) = x^2+980/x
S(x) = x^2+980x^(-1)
S'(x) = 2x-980x^(-2)
S'(x) = 2x-980/(x^2)
S'(x) = (2x^3)/(x^2)-980/(x^2)
S'(x) = (2x^3-980)/(x^2)
0 = (2x^3-980)/(x^2)
2x^3-980 = 0
2x^3 = 980
x^3 = 980/2
x^3 = 490
x = 490^(1/3)
x = 7.883735

The critical value is roughly x = 7.883735

Use the first derivative test
 Plug in x = 7
S'(x) = (2x^3-980)/(x^2)
S'(7) = (2(7)^3-980)/(7^2)
S'(7) = -6
then plug in x = 8
S'(x) = (2x^3-980)/(x^2)
S'(8) = (2*8^3-980)/(8^2)
S'(8) = 0.6875
The derivative changes sign as you move through the critical value. Since we change from negative to positive, this shows we have a local min at roughly x = 7.883735

Plug this x value into the h(x) function to get the height
h(x) = 245/(x^2)
h(7.883735) = 245/((7.883735)^2)
h(7.883735) = 3.941867
which rounds to 3.942

So you have the correct answer. Nice work getting two in a row.