Question

Rework Problem 37 if the belt is crossed, as shown in the figure.

Short answer

The length of the belt is$79.69inches.$

Step-by-step solution

Step 1. Given Information

Here, we can form two right triangles namely $∆ACI$and $∆BFI$.

To solve the length of the belt required to join the two, we only solve the measure of the lines and arcs in red as shown below, and multiply resulting value by 2.

Hence, $Belt=2\left(CI+FI+m\stackrel{⏜}{CH}+m\stackrel{⏜}{FG}\right)$

Step 2. Finding BI and AI

Let $BI=xandAI=24-x$

since the two angles of right triangles are congruent, then they are similar. Therefore,

$$\begin{gathered} \frac{x}{24-x}=\frac{2.5}{6.5} \\ 6.5x=60-2.5x \\ 9x=60 \\ x=\frac{20}{3} \end{gathered}$$

then$AI=24-\frac{20}{3}=\frac{52}{3}$

Step 3. Find CI and FI

Using Pythagoras Theorem,

$$\begin{gathered} C{I}^2=A{I}^2-A{C}^2 \\ CI=\sqrt{{\left(\frac{52}{3}\right)}^2-{\left(6.5\right)}^2} \end{gathered}$$

$CI=\sqrt{\frac{9295}{36}}=16.07$

and

$$\begin{gathered} F{I}^2=B{I}^2-B{F}^2 \\ FI=\sqrt{{\left(\frac{20}{3}\right)}^2-{\left(2.5\right)}^2} \\ FI=\sqrt{\frac{1375}{36}}=6.18 \end{gathered}$$

Step 4. Finding ∠CAI and ∠FBI

Since $∆ACI$and $∆FBI$are similar. So $\angle ACI\approx \angle FBI$

By using law of sines

$\frac{\sin A}{a}=\frac{\sin B}{b}$

$$\begin{gathered} \frac{\sin \angle FBI}{FI}=\frac{\sin 90°}{BI} \\ \sin \angle FBI=FI\times \frac{\sin 90°}{BI} \\ =\frac{6.18\times \sin 90°}{{\displaystyle \raisebox{1ex}{$20$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}=0.927 \\ \angle FBI={\sin }^{-1}\left(0.927\right)=67.98° \end{gathered}$$

Thus,$\angle ACI=\angle FBI=67.98°$

Step 5. Finding mCH⏜ and mFG⏜

$$\begin{gathered} m\stackrel{⏜}{CH}=2\mathrm{\pi r}\frac{180°-\angle \mathrm{CAI}}{360°} \\ =2\mathrm{\pi }\times 6.5\frac{180°-67.98°}{360°} \\ =12.71 \end{gathered}$$

and

$$\begin{gathered} m\stackrel{⏜}{FG}=2\mathrm{\pi r}\frac{180°-\angle FBI}{360°} \\ =2\mathrm{\pi }\times 2.5\frac{180°-67.98°}{360°} \\ =4.89 \end{gathered}$$

Step 6. Length of Belts

$$\begin{gathered} Belt=2\left(CI+FI+m\stackrel{⏜}{CH}+m\stackrel{⏜}{FG}\right) \\ =2\left(16.07+6.18+12.71+4.89\right)=79.69 \end{gathered}$$