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

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 =  

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 ∝

This can then be written as;

x = k()

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 ∝ ()

This can then be written as;

x = k ()

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 ∝

=> T² =  k ()        -------------------(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 ()

16 =  k ()

8k = 16 x 9

8k = 144

k =

k = 18

Now substitute the value of k back into equation (i);

T² =  18 ()

T² =  

Therefore, the general formula that describes the variation is;

T² =  

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step-by-step explanation: