Certainly! Let's break down the problem step-by-step:
### Problem Analysis
We are given two quantities, [tex]\( y \)[/tex] and [tex]\( x \)[/tex], which are in direct proportion. This means that the ratio between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] is constant. In mathematical terms, [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is the constant of proportionality.
Additionally, we are provided with information that the difference in the values of [tex]\( y \)[/tex] when [tex]\( x = 3 \)[/tex] and [tex]\( x = 8 \)[/tex] is 20.
### Solution
#### (a) Find [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
1. Express [tex]\( y \)[/tex] using the proportionality constant [tex]\( k \)[/tex]:
Since [tex]\( y \)[/tex] and [tex]\( x \)[/tex] are directly proportional, we can write:
[tex]\[
y = kx
\][/tex]
2. Use the given difference to find [tex]\( k \)[/tex]:
We know that the difference in the values of [tex]\( y \)[/tex] when [tex]\( x = 3 \)[/tex] and [tex]\( x = 8 \)[/tex] is 20.
So,
[tex]\[
y \text{ when } x = 8 - y \text{ when } x = 3 = 20
\][/tex]
Substituting [tex]\( y = kx \)[/tex]:
[tex]\[
k \cdot 8 - k \cdot 3 = 20
\][/tex]
3. Simplify and solve for [tex]\( k \)[/tex]:
[tex]\[
8k - 3k = 20
\][/tex]
[tex]\[
5k = 20
\][/tex]
[tex]\[
k = \frac{20}{5} = 4
\][/tex]
4. Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
Now that we know [tex]\( k = 4 \)[/tex], we can write:
[tex]\[
y = 4x
\][/tex]
#### (b) Find the value of [tex]\( y \)[/tex] when [tex]\( x = 10 \)[/tex]:
1. Substitute [tex]\( x = 10 \)[/tex] into the equation [tex]\( y = 4x \)[/tex]:
[tex]\[
y = 4 \cdot 10
\][/tex]
2. Calculate the value of [tex]\( y \)[/tex]:
[tex]\[
y = 40
\][/tex]
### Conclusion
(a) The relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] is:
[tex]\[
y = 4x
\][/tex]
(b) When [tex]\( x = 10 \)[/tex], the value of [tex]\( y \)[/tex] is:
[tex]\[
y = 40
\][/tex]
