User:Volvagia356/Sandbox

Description of Parabola
$$f(x)=ax^2+bx+c\,$$

$$f(x)=ax^2+bx+c$$

$$f(x)=a(x-p)^2+q$$

$$x+p=0$$

$$x=-p$$

$$\frac{b\pm\sqrt{b^2-4ac}}{2a}$$

First Function
$$4=a(-2)^2+b(-2)+4.5$$

$$4=4a-2b+4.5$$

$$4=4a+4.5$$

$$4a=-\frac{1}{2}$$

$$a=-\frac{1}{8}$$

$$y=-\frac{1}{8}x^2+4.5$$

Second Function
$$\frac{1}{2}$$

$$\frac{1}{2}=0a^2+0b+c$$

$$c=\frac{1}{2}$$

$$y=-\frac{1}{8}x^2+\frac{1}{2}$$

Third Function
$$-\frac{1}{8}(x-2)^2+\frac{1}{2}$$

Finding the Surface Area
$$\int_{-2}^{2}-\frac{1}{8}x^2+\frac{1}{2}\, dx$$

$$=\int_{-2}^{2}\frac{1}{2}\, dx-\frac{1}{8}\int_{-2}^{2}x^2\, dx$$

$$=\int_{-2}^{2}\frac{1}{2}\, dx-\left [\frac{x^3}{24}\right]_{-2}^2$$

$$=\left [\frac{x}{2}-\frac{x^3}{24}+c\right]_{-2}^2$$

integral of $$x^2$$ is $$\frac{x^3}{3}$$

integral of 1/2 is $$\frac{x}{2}$$

$$=\left (\frac{2}{2}-\frac{2^3}{24}+c\right)-\left (\frac{-2}{2}-\frac{-2^3}{24}+c\right)$$

$$=\frac{4}{3}$$

$$=\frac{4}{3}\operatorname{m}^2$$

Volume & Cost of Arch
Structure 1

$$4\operatorname{m}^2-\frac{4}{3}\operatorname{m}^2=\frac{8}{3}\operatorname{m}^2$$

$$\frac{8}{3}\operatorname{m}^2\times0.4\operatorname{m}=\frac{16}{15}\operatorname{m}^3$$

$$\frac{16}{15}\operatorname{m}^3\times\operatorname{RM}840=\operatorname{RM}896$$

Structure 2

$$1\operatorname{m}\times 4\operatorname{m}=4\operatorname{m}^2$$

$$\frac{1}{2}\times0.5\operatorname{m}\times4\operatorname{m}=1\operatorname{m}^2$$

$$4\operatorname{m}^2-1\operatorname{m}^2=3\operatorname{m}^2$$

$$3\operatorname{m}^2\times0.4\operatorname{m}=\frac{6}{5}\operatorname{cm}^3$$

$$\frac{6}{5}\operatorname{m}^3\times\operatorname{RM}840=\operatorname{RM}1008$$

Structure 3

$$\frac{1}{2}\times\left(1\operatorname{m}+4\operatorname{m}\right)\times0.5\operatorname{m}=\frac{5}{4}\operatorname{m}^2$$

$$4\operatorname{m}^2-\frac{5}{4}\operatorname{m}^2=\frac{11}{4}\operatorname{m}^2$$

$$\frac{11}{4}\operatorname{m}^2\times0.4\operatorname{m}=\frac{11}{20}\operatorname{m}^3$$

$$\frac{11}{20}\operatorname{m}^3\times\operatorname{RM}840=\operatorname{RM}924$$

Structure 4

$$\frac{1}{2}\times\left(2\operatorname{m}+4\operatorname{m}\right)\times0.5\operatorname{m}=\frac{3}{2}\operatorname{m}^2$$

$$4\operatorname{m}^2-\frac{3}{2}\operatorname{m}^2=\frac{5}{2}\operatorname{m}^2$$

$$\frac{5}{2}\operatorname{m}^2\times0.4\operatorname{m}=1\operatorname{m}^3$$

$$1\operatorname{m}^3\times\operatorname{RM}840=\operatorname{RM}840$$

Different Sizes
$$4\times1-\frac{0+4}{2}\times0.5=3$$

$$4\times1-\frac{0.25+4}{2}\times0.5=2.9375$$

$$4\times1-\frac{0.5+4}{2}\times0.5=2.875$$

$$4\times1-\frac{0.75+4}{2}\times0.5=2.8125$$

$$4\times1-\frac{1+4}{2}\times0.5=2.75$$

$$4\times1-\frac{1.25+4}{2}\times0.5=2.6875$$

$$4\times1-\frac{1.5+4}{2}\times0.5=2.625$$

$$4\times1-\frac{1.75+4}{2}\times0.5=2.5625$$

$$4\times1-\frac{2+4}{2}\times0.5=2.5$$

$$=4\times1-\frac{k+4}{2}\times0.5$$

$$=4-\left(\frac{k}{4}+1\right)$$

$$=3-\frac{k}{4}$$

$$\operatorname{as}\ k\rightarrow4,\ \frac{k}{4}\rightarrow1$$