To determine the relationship and find \( y \) in terms of \( x \), let's break down the problem step-by-step.
1. Identify Relationships:
- Given that \( y \) is directly proportional to \( w^2 \), we can write:
[tex]\[
y = k_1 w^2
\][/tex]
where \( k_1 \) is the proportionality constant.
- Provided that \( x \) is inversely proportional to \( w \), we can write:
[tex]\[
x = \frac{k_2}{w}
\][/tex]
where \( k_2 \) is the proportionality constant for this relationship.
2. Determine the Proportionality Constants:
- From \( y = k_1 w^2 \):
[tex]\[
y = 5 \quad \text{when} \quad w = 10
\][/tex]
Substituting the values, we get:
[tex]\[
5 = k_1 \times 10^2 \quad \Rightarrow \quad 5 = 100 k_1 \quad \Rightarrow \quad k_1 = \frac{5}{100} = 0.05
\][/tex]
- From \( x = \frac{k_2}{w} \):
[tex]\[
x = 0.4 \quad \text{when} \quad w = 10
\][/tex]
Substituting the values, we get:
[tex]\[
0.4 = \frac{k_2}{10} \quad \Rightarrow \quad k_2 = 0.4 \times 10 = 4.0
\][/tex]
3. Express \( y \) in Terms of \( x \):
- First, express \( w \) in terms of \( x \):
[tex]\[
w = \frac{k_2}{x}
\][/tex]
- Now substitute this \( w \) into the expression for \( y \):
[tex]\[
y = k_1 w^2 = k_1 \left( \frac{k_2}{x} \right)^2
\][/tex]
- Simplify the expression:
[tex]\[
y = k_1 \frac{k_2^2}{x^2}
\][/tex]
- Substitute the values of \( k_1 \) and \( k_2 \):
[tex]\[
y = 0.05 \frac{4.0^2}{x^2} = 0.05 \frac{16}{x^2}
\][/tex]
- Finally, simplify the expression for the relationship:
[tex]\[
y = \frac{0.8}{x^2}
\][/tex]
Thus, the expression for \( y \) in terms of \( x \) in its simplest form is:
[tex]\[
y = \frac{0.8}{x^2}
\][/tex]
