Question

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

Answer 1

Question:

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

Answer:

T² =  $\frac{18a^3}{d^2}$

Step-by-step explanation:

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 ∝ $\frac{1}{y}$

This can then be written as;

x = k($\frac{1}{y}$)

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 ∝ ($\frac{y}{z}$)

This can then be written as;

x = k ($\frac{y}{z}$)

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 ∝ $\frac{a^3}{d^2}$

=> T² =  k ($\frac{a^3}{d^2}$)        -------------------(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 ($\frac{2^3}{3^2}$)

16 =  k ($\frac{8}{9}$)

8k = 16 x 9

8k = 144

k = $\frac{144}{8}$

k = 18

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

T² =  18 ($\frac{a^3}{d^2}$)

T² =  $\frac{18a^3}{d^2}$

Therefore, the general formula that describes the variation is;

T² =  $\frac{18a^3}{d^2}$