#### Answer

The minimum diameter of the hose is 3.5 cm

#### Work Step by Step

The pressure difference between the pressure inside the ideal vacuum cleaner and the atmospheric pressure is $1.01\times 10^5~N/m^2$. The force exerted on the dog is equal to this pressure difference multiplied by the area $A$ of the hose. To find the minimum radius of the hose, we can let this force equal the dog's weight. Therefore;
$F = (1.01\times 10^5~N/m^2)~A = Mg$
$(1.01\times 10^5~N/m^2)~(\pi~R^2) = Mg$
$R^2 = \frac{Mg}{(1.01\times 10^5~N/m^2)~(\pi)}$
$R = \sqrt{\frac{Mg}{(1.01\times 10^5~N/m^2)~(\pi)}}$
$R = \sqrt{\frac{(10~kg)(9.80~m/s^2)}{(1.01\times 10^5~N/m^2)~(\pi)}}$
$R = 0.01757~m = 1.757~cm$
We then find the minimum diameter of the hose.
$d = 2R = (2)(1.757~cm) = 3.5~cm$
Therefore, the minimum diameter of the hose is 3.5 cm.