Rom

I'm seeing (5,20), (6, 12), (8,8) as the possible cool rectangles so 

100 + 72 + 64 = 236

i'm gonna assume you can plug p=3/4 in

correct though you should say it's farther away from the center, or from (0,0)

\(\text{let }P[\text{heads}] = p\\ E[V] = 2p -(1-p) = 3p-1\)

I suspect there were numbers for p and (1-p) but they didn't come through.

The idea is to factor the stuff under the radical into two pieces.

One piece is something that has a clear cube or 4th root and the other piece is the rest.

In the first one we want to break \(-54 n^7\) into pieces one of which we can simply take the cube root of.

\(\left(-3n^2\right)^3 = -27n^6\\ -54n^7 = -27n^6 \cdot 2n \\ \sqrt[3]{-54n^7} = \sqrt[3]{-27n^6 \cdot 2n} = -3n^2 \sqrt[3]{2n} \\ -\sqrt[3]{-54n^7} = -(-3n^2\sqrt[3]{2n}) = 3n^2\sqrt[3]{2n} \)

I'll do one more for you but I'd like to see you work these out yourself

\(-5\sqrt[4]{128x^4} = -5\sqrt[4]{2^7 x^4}=\\ -5\sqrt[4]{2^4 x^4 \cdot 2^3} = -5\cdot 2x\sqrt[4]{2^3} = \\ -10x \sqrt[4]{8}\)

\(-\sqrt[3]{-54n^7} = \\ -((-1)\sqrt[3]{54n^7} = \\ \sqrt[3]{27n^6 \cdot 2n} = \\ 3n^2\sqrt[3]{2n}\)

the rest are similar, give them another try

\(\text{well the domain of y(x) is }[-2,2] \\ \text{and over this domain y increases from 0 to }\sqrt{2} \text{ and then decreases back to 0}\\ \text{so the range of y(x) is}\\ \left\{y:y \in [0,\sqrt{2}]\right\}\\ \text{what you've written isn't quite correct. y(x) never gets above }y=\sqrt{2}\)

\(\dfrac{c}{t} = \dfrac{7}{3} \\ \dfrac{t+600}{t}=\dfrac 7 3 \\ 3t+1800 = 7t \\ 4t = 1800 \\ t=450\)

\(c(n) = fixedcost + n \cdot percopycost\\ 44 = fixedcost+70 \cdot percopycost \\ 62 = fixedcost + 231 \cdot percopycost\\ \text{subtract the first equation from the second}\\ 18= 161 \cdot percopycost \\ percopycost=\dfrac{18}{161} \approx $0.112/copy\\ fixedcost = 44-\dfrac{70\cdot 18}{161} \approx $36.174\\ c(n) = $(36.174 + 0.112n)\)