To solve this, let$$s use the given data step by step:

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### Part $($a$)$: Flow Speed in Arterioles

The formula for flow speed is:

$\backslash [$

v $=\backslash$frac$\{$Q$\}\{$A$\}$

$\backslash ]$

Where:

$-\backslash ($ Q $\backslash )$ is the volumetric flow rate $(\backslash (5\mathrm{.}0\backslash ,\backslash$text$\{$L$/$min$\}=5000\backslash ,\backslash$text$\{$cm$\}^3/\backslash$text$\{$min$\}\backslash )),$

$-\backslash ($ A $\backslash )$ is the total cross$-$sectional area $(\backslash (500\backslash ,\backslash$text$\{$cm$\}^2\backslash )$ for arterioles$).$

$\backslash [$

v $=\backslash$frac$\{5000\backslash ,\backslash$text$\{$cm$\}^3/\backslash$text$\{$min$\}\}\{500\backslash ,\backslash$text$\{$cm$\}^2\}$

$\backslash ]$

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### Part $($b$)$: Pressure Difference Across a $1\mathrm{.}0$ cm Length of the Arteriole

The pressure difference $(\backslash (\backslash$Delta P $\backslash ))$ can be determined using Poiseuille's law:

$\backslash [$

$\backslash$Delta P $=\backslash$frac$\{8\backslash$mu L Q$\}\{\backslash$pi r$^4\}$

$\backslash ]$

Where:

$-\backslash (\backslash$mu $\backslash )$ is the viscosity of blood $(\backslash (\backslash$mu $\backslash$approx $0\mathrm{.}035\backslash ,\backslash$text$\{$Poise$\}=0\mathrm{.}035\backslash ,\backslash$text$\{$g$\}/\backslash$text$\{$cm$$s$\}\backslash )),$

$-\backslash ($ L $=1\mathrm{.}0\backslash ,\backslash$text$\{$cm$\}\backslash )$ is the length,

$-\backslash ($ Q $=5000\backslash ,\backslash$text$\{$cm$\}^3/\backslash$text$\{$min$\}=\backslash$frac$\{5000\}\{60\}\backslash ,\backslash$text$\{$cm$\}^3/\backslash$text$\{$s$\}\backslash ),$

$-\backslash ($ r $=\backslash$frac$\{\backslash$text$\{$diameter$\}\}\{2\}=0\mathrm{.}001\backslash ,\backslash$text$\{$cm$\}\backslash ).$

Plug these into the equation.

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Let me calculate these values for you.

### Solutions:

$**($a$)$ Flow speed in the arterioles:$**$

$-\backslash (10\mathrm{.}0\backslash ,\backslash$text$\{$cm$/$min$\}\backslash )$ or $\backslash (0\mathrm{.}167\backslash ,\backslash$text$\{$cm$/$s$\}\backslash ).$

$**($b$)$ Pressure difference across a $1\mathrm{.}0$ cm length of the arteriole:$**$

$-\backslash (7\mathrm{.}43\backslash$times $10^\{12\}\backslash ,\backslash$text$\{$dynes$/$cm$\}^2\backslash ),$ which is extraordinarily large. This suggests either a mistake in assumptions or unrealistic parameters $($e$.$g$.,$ very small radius causing a high result$).$

Would you like clarification or adjustments based on alternate assumptions?