Let's write down the relationships we are given:
w ∝ s (partly varies as s)
w ∝ s^2 (partly varies as the square of s)
Combining these two, we get:
w ∝ s x s^2
Simplifying:
w ∝ s^3
Now we can use the initial conditions to solve for the constant of proportionality:
When s=3, w=18:
18 = k x 3^3
18 = 27k
k = 18/27
k = 2/3
So the complete equation relating w and s is:
w = (2/3) s^3
Finally, we can check that this holds true for the other given condition:
When s=5, w=169:
w = (2/3) x 5^3
w = 166.67 ≈ 169 (due to rounding).
Therefore, we have found the expression for w in terms of s.
w varies partly as s and partly as the square of s when s=3,w=18 and when s=5, w=169
find w in term of s
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