Answer :
To find the percentage change in the variable y when x is changed in the ratio of 5:4 and z is reduced by 19%, we will follow the steps below:
Step 1: Understand the initial relationship
The initial relation says that y is directly proportional to \( x^2 \) and inversely proportional to \( \sqrt{z} \). This can be expressed mathematically as:
\[ y \propto x^2 \]
\[ y \propto \frac{1}{\sqrt{z}} \]
These relationships can be combined as:
\[ y = k \cdot \frac{x^2}{\sqrt{z}} \]
where \( k \) is a constant of proportionality.
Step 2: Change in y due to change in x
x is changed in the ratio 5:4, which means if x originally was 4 units, it now becomes 5 units. The change in y due to change in x can be calculated based on the fact that y varies as the square of x:
The new value of y due to the change in x (before considering changes in z) can be found as:
\[ y_{new\_x} = k \cdot \left(\frac{5}{4}\right)^2 \cdot \frac{1}{\sqrt{z}} \]
\[ y_{new\_x} = k \cdot \frac{25}{16} \cdot \frac{1}{\sqrt{z}} \]
The ratio of the new y due to change in x to the original y is therefore \( \frac{25}{16} \).
Step 3: Change in y due to change in z
z is reduced by 19%. If z originally is 100% of some value, it will now be 81% (which is 100% - 19%) of that value. To find the change in y due to the change in z, we'll account for the inverse relationship of y with \( \sqrt{z} \):
The new value of z is:
\[ z_{new} = 0.81 \cdot z \]
The new value of y due to the change in z (before considering changes in x) can be found as:
\[ y_{new\_z} = k \cdot \frac{x^2}{\sqrt{0.81 \cdot z}} \]
Recognize that \( \sqrt{0.81} \) is the same as \( 0.9 \) (since \( 0.9^2 = 0.81 \)). The relationship can then be rewritten as:
\[ y_{new\_z} = k \cdot \frac{x^2}{0.9 \cdot \sqrt{z}} \]
So the new y due to z becoming 81% of its original value is \( \frac{1}{0.9} \) times greater than the original y.
Step 4: Combine both changes to get the overall change
Because of the direct and inverse relationships, these changes multiply. The overall factor by which y changes is the product of individual changes due to x and z:
\[ \text{Overall change factor} = \frac{25}{16} \cdot \frac{1}{0.9} \]
Step 5: Calculate the percentage change
To calculate the percentage change, we multiply the overall change factor by 100 and subtract 100:
\[ \text{Percentage change in y} = \left(\frac{25}{16} \cdot \frac{1}{0.9} - 1\right) \cdot 100\% \]
Now we can plug in the values to get the answer:
\[ \text{Percentage change in y} = \left(\frac{25}{16} \cdot \frac{10}{9} - 1\right) \cdot 100\% \]
\[ \text{Percentage change in y} = \left(\frac{250}{144} - 1\right) \cdot 100\% \]
\[ \text{Percentage change in y} = \left(\frac{106}{144}\right) \cdot 100\% \]
\[ \text{Percentage change in y} = \frac{106}{1.44}\% \]
\[ \text{Percentage change in y} \approx 73.61\% \]
Therefore, variable y increases by approximately 73.61% when x is changed in the ratio 5:4 and z is reduced by 19%.
