If x varies directly as square root of y and x = 81 when y = 9, Find x when y = 1$\frac{7}{9}$

1. Analyze the question

We are given a variation problem where $x$ varies directly as the square root of $y$. We are provided with an initial set of values ($x=81, y=9$) to find the constant of variation, and then asked to find the new value of $x$ when $y = 1\frac{7}{9}$.

2. Explain the concept

• Direct Variation: When $x$ varies directly as $\sqrt{y}$, it is written mathematically as $x \propto \sqrt{y}$.
• Constant of Variation: To turn this into an equation, we introduce a constant $k$: $$x = k\sqrt{y}$$
• Steps: • Find $k$ using the first set of values. • Use the value of $k$ to solve for $x$ using the second value of $y$.

3. Show the solution

Step 1: Find the constant $k$ Given $x = 81$ and $y = 9$:

Step 2: Find $x$ when $y = 1\frac{7}{9}$ First, convert the mixed fraction to an improper fraction:

Now, substitute $k = 27$ and $y = \frac{16}{9}$ into the equation $x = k\sqrt{y}$:

4. Identify the answer

Comparing our result to the options provided: A. 20$\frac{1}{4}$ B. 27 C. 2$\frac{1}{4}$ D. 36

The correct option is D.

5. Tip

When dealing with square roots in variation problems, always convert mixed fractions to improper fractions first. It makes finding the square root much easier because the numerator and denominator will often be perfect squares (like 16 and 9 in this case).