Q. 74 - Calculus

Question

Dr. Geek built a skate ramp in his backyard modeled after the function $f\left(x\right)=11 \sin \frac{1}{42}{x}^2$, measured in feet, as shown in the next figure. Set up a definite integral that measures the distance required to skate from one side of the ramp to the other. The starting and ending x-values should both satisfy $f\left(x\right)=11 feet$. Use a graphing calculator to approximate the value of this definite integral.

Step-by-Step Solution

Answer

Distance required to skate from one side of the ramp to the other is: $28.3804 feet$. 

Step 1. Given information

Skate ramp modeled after the function, $f\left(x\right)=11 \sin \frac{1}{42}{x}^2$

Step 2. Finding x   when   f ( x ) = 11   f e e t

Skate ramp curve is: $f\left(x\right)=11 \sin \frac{1}{42}{x}^2$

Put, $f\left(x\right)=11 \mathrm{and} \mathrm{find} x$

$$\begin{gathered} f\left(x\right)=11 \sin \frac{1}{42}{x}^2=11 \\ \Rightarrow \sin \frac{1}{42}{x}^2=1 \\ \Rightarrow \frac{x^2}{42}={\sin }^{-1}1 \\ \Rightarrow x^2=42\times \frac{\mathrm{\pi }}{2} \\ \Rightarrow x^2=21\times 3.1415 \\ \Rightarrow x=\sqrt{65.9734} \\ \Rightarrow x=\pm 8.1224 \end{gathered}$$

Step 3. Finding distance required to skate from one side of the ramp to the other.

$$\begin{gathered} \mathrm{The} \mathrm{arc} \mathrm{length} \mathrm{of} f\left(x\right) \mathrm{from} x=a \mathrm{to} x=b \mathrm{can} \mathrm{be} \mathrm{represented} \mathrm{by} \mathrm{the} \\ \mathrm{definite} \mathrm{integral}:{\int }_{\mathrm{a}}^{\mathrm{b}}\sqrt{1 + ( \mathrm{f}\text{'}{\left(\mathrm{x}\right))}^2}d\mathrm{x} \\ \mathrm{Equation} \mathrm{of} \mathrm{skate} \mathrm{ramp} \mathrm{curve} \mathrm{is}: \mathrm{f}\left(\mathrm{x}\right)=11 \sin \frac{1}{42}{\mathrm{x}}^2 \\ \mathrm{and} \mathrm{f}\text{'}\left(\mathrm{x}\right)=\frac{11 \times 2\mathrm{x}}{42} \cos \frac{1}{42}{\mathrm{x}}^2=\frac{11\mathrm{x}}{21}\cos \frac{1}{42}{\mathrm{x}}^2 \\ \mathrm{Distance} \mathrm{required} \mathrm{to} \mathrm{skate} \mathrm{from} \mathrm{one} \mathrm{side} \mathrm{of} \mathrm{the} \mathrm{ramp} \mathrm{to} \mathrm{the} \mathrm{other} \mathrm{is} \mathrm{given} \mathrm{by}, \\ \mathrm{D}={\int }_{-8.1224}^{8.1224}\sqrt{1 + {\left(\frac{11\mathrm{x}}{21}\cos \frac{1}{42}{\mathrm{x}}^2\right)}^2}d\mathrm{x} \\ \mathrm{Using} \mathrm{graphing} \mathrm{calculator}, \mathrm{the} \mathrm{approximate} \mathrm{value} \mathrm{of} \mathrm{this} \mathrm{definite} \mathrm{integral} \mathrm{is}: \\ \mathrm{D}=28.3804 \end{gathered}$$