A variable $$ P $$ varies jointly with the square and inversely with the square root of $$ Q $$. If ncreased by $$ 10 \% $$ and $$ Q $$ decreased by $$ 20 \% $$, wh e percentage change in the value of $$ P $$ ?

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Let's break down the problem step by step.

**Given:**
- $$ P $$ varies jointly with the square of $$ n $$ and inversely with the square root of $$ Q $$.
- $$ n $$ is increased by $$ 10\% $$.
- $$ Q $$ is decreased by $$ 20\% $$.

**Mathematical Representation:**
$$ P = k \cdot \frac{n^2}{\sqrt{Q}} $$
where $$ k $$ is the constant of proportionality.

**Step 1: Adjust for Change in $$ n $$**
- $$ n $$ is increased by $$ 10\% $$:
$$ n_{\text{new}} = n \times 1.10 $$

**Step 2: Adjust for Change in $$ Q $$**
- $$ Q $$ is decreased by $$ 20\% $$:
$$ Q_{\text{new}} = Q \times 0.80 $$

**Step 3: Substitute the New Values into $$ P $$**
$$ P_{\text{new}} = k \cdot \frac{(1.10n)^2}{\sqrt{0.80Q}} $$
$$ P_{\text{new}} = k \cdot \frac{1.21n^2}{\sqrt{0.80} \cdot \sqrt{Q}} $$
$$ P_{\text{new}} = \frac{1.21}{\sqrt{0.80}} \cdot \frac{k n^2}{\sqrt{Q}} $$
$$ P_{\text{new}} = \frac{1.21}{0.894427} \cdot P $$
$$ P_{\text{new}} \approx 1.3535 \cdot P $$

**Step 4: Calculate the Percentage Change**
$$ \text{Percentage Change} = (1.3535 - 1) \times 100\% $$
$$ \text{Percentage Change} \approx 35.35\% $$

**Conclusion:**
The value of $$ P $$ increases by approximately **35%**.

**Answer:**  
The value of $$ P $$ increases by approximately 35 %.
The value of $$ P $$ increases by approximately 35%.

A variable varies jointly with the square
and inversely with the square root of . If
ncreased by and decreased by , wh
e percentage change in the value of ?

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A variable varies jointly with the square and inversely with the square root of . If ncreased by and decreased by , wh e percentage change in the value of ?