本题目来源于试卷: Integral Calculus，类别为 IB数学
[问答题]
Consider the functioxqk3-i2rx1 2o tmzji **z vnm1 ) h6c7rz i4frdy.bl)wtr;kx5n $f(x)=\sqrt{\frac{10}{x^{2}}-1}$ , where $1 \leq x \leq \sqrt{10} $.
1. Sketch the curve y=f(x) , indicating the coordinates of the endpoints.
2. 1. Show that $f^{-1}(x)=\sqrt{\frac{10}{x^{2}+1}}$.
2. State the domain and range of $ f^{-1}$ .
The curve y=f(x) is rotated through $2 \pi$ about the y -axis to form a solid of revolution that is used to model a vase.
3. 1. Show that the volume $ V \mathrm{~cm}^{3}$ , of liquid in the vase when it is filled to a height of h centimetres is given by $V=10 \pi \arctan (h)$ .
2. Hence, determine the volume of the vase.
At t=0 , the vase is filled to its maximum volume with water. Water is then removed from the vase at a constant rate of $4 \mathrm{~cm}^{3} \mathrm{~s}^{-1}$ .
4. Find the time it takes to completely empty the vase.
5. Find the rate of change of the height of the water when half of the water has been emptied from the vase.
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