Solving a system of equations with inequalities and trigoniometry

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Question, find the upper and lower bounds of $\text{P}$,$\text{Q}$ and $\text{S}$ when:

  1. $$ \begin{cases} \text{P}=\text{U}_{\text{Lf}}\cdot\text{I}_{\text{Lf}}\cdot\sqrt{3}\cdot\cos\left(\varphi\right)\\ \text{Q}=\text{U}_{\text{Lf}}\cdot\text{I}_{\text{Lf}}\cdot\sqrt{3}\cdot\sin\left(\varphi\right)\\ \text{S}=\sqrt{\text{P}^2+\text{Q}^2}\\ 0<\varphi\le\arccos\left(\frac{17}{20}\right)\\ \text{U}_{\text{Lf}}=230\cdot\sqrt{3}\\ 0<\text{I}_{\text{Lf}}\le10 \end{cases} $$

  2. $$ \begin{cases} \text{P}=\text{U}_{\text{Lf}}\cdot\text{I}_{\text{Lf}}\cdot\sqrt{3}\cdot\cos\left(\varphi\right)\\ \text{Q}=\text{U}_{\text{Lf}}\cdot\text{I}_{\text{Lf}}\cdot\sqrt{3}\cdot\sin\left(\varphi\right)\\ \text{S}=\sqrt{\text{P}^2+\text{Q}^2}\\ 0<\varphi\le\arccos\left(\frac{17}{20}\right)\\ \text{U}_{\text{Lf}}=230\\ 0<\text{I}_{\text{Lf}}\le16 \end{cases} $$


My work, when I knew $\text{P}=7000$ instead of the boundaries of $\text{I}_{\text{Lf}}$, I solved it this way:

$$ \begin{cases} \text{P}=\text{U}_{\text{Lf}}\cdot\text{I}_{\text{Lf}}\cdot\sqrt{3}\cdot\cos\left(\varphi\right)\\ \text{Q}=\text{U}_{\text{Lf}}\cdot\text{I}_{\text{Lf}}\cdot\sqrt{3}\cdot\sin\left(\varphi\right)\\ \text{S}=\sqrt{\text{P}^2+\text{Q}^2}\\ 0<\varphi\le\arccos\left(\frac{17}{20}\right)\\ \text{U}_{\text{Lf}}=230\cdot\sqrt{3}\\ \text{P}=7000 \end{cases}\therefore \begin{cases} \frac{700}{69}<\text{I}_{\text{Lf}}\le\frac{14000}{1173}\\ 0<\text{Q}\le\frac{7000\sqrt{111}}{17}\\ 7000<\text{S}\le\frac{140000}{17} \end{cases}$$

But for the ones above I do not get it.