Write a general formula to describe each variation.
The square of T varies directly with the cube of a and inversely with the square of d; T = 4 when a = 2 and d = 3
T² = [tex]\frac{18a^3}{d^2}[/tex]
Few things to note:
i. direct variation: When a variable x varies directly with another variable y, we write it in this form;
x ∝ y.
This can then be written as;
x = ky
Where;
k = constant of proportionality variation.
ii. inverse variation: When a variable x varies inversely with another variable y, we write it in this form;
x ∝ [tex]\frac{1}{y}[/tex]
This can then be written as;
x = k([tex]\frac{1}{y}[/tex])
Where;
k = constant of proportionality or variation
iii. combined variation: When a variable x varies directly with variable y and inversely with variable z, we write it in this form;
x ∝ ([tex]\frac{y}{z}[/tex])
This can then be written as;
x = k ([tex]\frac{y}{z}[/tex])
Where;
k = constant of proportionality or variation
From the question;
The square of T varies directly with the cube of a and inversely with the square of d.
Note that
square of T = T²
cube of a = a³
square of d = d²
Therefore, we can write;
T² ∝ [tex]\frac{a^3}{d^2}[/tex]
=> T² = k ([tex]\frac{a^3}{d^2}[/tex]) -------------------(i)
Since;
T = 4 when a = 2 and d = 3
We can find the constant of proportionality k, by substituting the values of T=4, a = 2 and d = 3 into equation (i) and solve as follows;
(4)² = k ([tex]\frac{2^3}{3^2}[/tex])
16 = k ([tex]\frac{8}{9}[/tex])
8k = 16 x 9
8k = 144
k = [tex]\frac{144}{8}[/tex]
k = 18
Now substitute the value of k back into equation (i);
T² = 18 ([tex]\frac{a^3}{d^2}[/tex])
T² = [tex]\frac{18a^3}{d^2}[/tex]
Therefore, the general formula that describes the variation is;
T² = [tex]\frac{18a^3}{d^2}[/tex]
