3. The deflection, $$ y $$, at the centre of a rod under constant load, varies directly as the cube of the length, $$ L $$, and indirectly as the fourth power of the diameter, $$ d $$, of the rod. Calculate the percentage change in $$ y $$ if $$ L $$ decreases by $$ 1 \% $$ and $$ d $$ increases by $$ 2 \% $$. (b) Make $$ R $$ the subject of the following formula:

Question

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### (a)
We are given that  
$$
y = k\frac{L^3}{d^4},
$$  
where $$ k $$ is a constant. When the length and diameter change, the new deflection is  
$$
y' = k\frac{(L')^3}{(d')^4}.
$$

Since the length $$ L $$ decreases by $$ 1\% $$, the new length is  
$$
L' = 0.99\,L.
$$
Similarly, since the diameter $$ d $$ increases by $$ 2\% $$, the new diameter is  
$$
d' = 1.02\,d.
$$

Substitute these into the expression for $$ y' $$:
$$
y' = k\frac{(0.99\,L)^3}{(1.02\,d)^4} 
= k\frac{0.99^3\,L^3}{1.02^4\,d^4}.
$$

Taking the ratio of the new deflection to the original deflection $$ y $$:
$$
\frac{y'}{y} = \frac{k\frac{0.99^3\,L^3}{1.02^4\,d^4}}{k\frac{L^3}{d^4}} 
= \frac{0.99^3}{1.02^4}.
$$

Now, we calculate the numerical values:
$$
0.99^3 \approx 0.970299, \qquad 1.02^4 \approx 1.082432.
$$
Thus,
$$
\frac{y'}{y} \approx \frac{0.970299}{1.082432} \approx 0.8965.
$$

This ratio means the deflection is about $$ 89.65\% $$ of its former value. The percentage change in $$ y $$ is then
$$
(0.8965 - 1) \times 100\% \approx -10.35\%.
$$
So, the deflection $$ y $$ decreases by approximately $$ 10.35\% $$.

### (b)
The problem statement for part (b) asks:  
“Make $$ R $$ the subject of the following formula:”  
However, the formula in which to isolate $$ R $$ has not been provided.

Please supply the complete formula so that we can rearrange it to make $$ R $$ the subject.
### (a) The deflection $$ y $$ decreases by approximately $$ 10.35\% $$. ### (b) Cannot determine $$ R $$ as the subject without the given formula.

The deflection, , at the centre of a rod under constant load, varies directly as the cube of
the length, , and indirectly as the fourth power of the diameter, , of the rod.
Calculate the percentage change in if decreases by and increases by .
(b) Make the subject of the following formula:

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(a)

(b)

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(a)

(b)

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The deflection, , at the centre of a rod under constant load, varies directly as the cube of the length, , and indirectly as the fourth power of the diameter, , of the rod. Calculate the percentage change in if decreases by and increases by . (b) Make the subject of the following formula: