wind resistance, measured in newtons, is modeled by f(x) = x³/3405000 - x²/1000 + 145097x/141875 + 1054…

Explanation:

Step1: Find the derivative of $f(x)$

We have $f(x)=\frac{x^{3}}{3405000}-\frac{x^{2}}{1000}+\frac{145097x}{141875}+ 1054$. Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we get $f^\prime(x)=\frac{3x^{2}}{3405000}-\frac{2x}{1000}+\frac{145097}{141875}$. Simplify $f^\prime(x)$ to $f^\prime(x)=\frac{x^{2}}{1135000}-\frac{x}{500}+\frac{145097}{141875}$.

Step2: Set $f^\prime(x) = 0$

To find the critical points, we set $\frac{x^{2}}{1135000}-\frac{x}{500}+\frac{145097}{141875}=0$. Multiply through by 1135000 to clear the fractions: $x^{2}-2270x + 932776=0$.

Step3: Solve the quadratic equation

Using the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for the quadratic equation $ax^{2}+bx + c = 0$. Here, $a = 1$, $b=-2270$, and $c = 932776$. Then $x=\frac{2270\pm\sqrt{(-2270)^{2}-4\times1\times932776}}{2\times1}=\frac{2270\pm\sqrt{5152900 - 3731104}}{2}=\frac{2270\pm\sqrt{1421796}}{2}=\frac{2270\pm1192.47}{2}$. We get two solutions: $x_1=\frac{2270 + 1192.47}{2}=1731.24$ and $x_2=\frac{2270-1192.47}{2}=538.77$. Both values are in the interval $[500,3100]$.

Step4: Evaluate $f(x)$ at critical points and endpoints

Evaluate $f(x)$ at $x = 500$, $x=538.77$, $x = 1731.24$ and $x = 3100$. $f(500)=\frac{500^{3}}{3405000}-\frac{500^{2}}{1000}+\frac{145097\times500}{141875}+1054$ $=\frac{125000000}{3405000}-250+\frac{72548500}{141875}+1054$ $\approx36.71-250 + 511.36+1054=1352.07$. $f(538.77)=\frac{(538.77)^{3}}{3405000}-\frac{(538.77)^{2}}{1000}+\frac{145097\times538.77}{141875}+1054$. $f(1731.24)=\frac{(1731.24)^{3}}{3405000}-\frac{(1731.24)^{2}}{1000}+\frac{145097\times1731.24}{141875}+1054$. $f(3100)=\frac{3100^{3}}{3405000}-\frac{3100^{2}}{1000}+\frac{145097\times3100}{141875}+1054$. After calculating, we find that the minimum value occurs at $x\approx538.77$ and $f(x)\approx1349.32$.

Answer:

Minimum wind resistance $f(x)=1349.32$ Newtons is obtained at $x = 538.77$ meters per second