Question Number 209630 by otchereabdullai@gmail.com last updated on 16/Jul/24
$$\:\:{If}\:{A}\:\:{varies}\:{as}\:{r}^{\mathrm{2}} \:{and}\:{V}\:\:{varies}\:{as}\:{r}^{\mathrm{3}} \\ $$$$\:{find}\:{percentage}\:{increase}\:{in}\:{A}\:{and}\:{V} \\ $$$$\:{if}\:\:{r}\:{is}\:{increased}\:{by}\:\mathrm{20\%} \\ $$
Answered by som(math1967) last updated on 17/Jul/24
![A∝r^2 ⇒A=kr^2 [k variation constant] A_(increase) =k(1.2r)^2 =1.44kr^2 % Increase in A=((1.44kr^2 −kr^2 )/(kr^2 ))×100 = ((.44kr^2 )/(kr^2 ))×100=44% V∝r^3 ⇒V=Kr^3 [K variation constant] V_(increase) =K×(1.2r)^3 =K×1.728r^3 % Increase=((1.728Kr^3 −Kr^3 )/(Kr^3 ))×100 = 72.8%](https://www.tinkutara.com/question/Q209632.png)
$$\:{A}\propto{r}^{\mathrm{2}} \:\Rightarrow{A}={kr}^{\mathrm{2}} \:\:\left[{k}\:{variation}\:{constant}\right] \\ $$$$\:{A}_{{increase}} ={k}\left(\mathrm{1}.\mathrm{2}{r}\right)^{\mathrm{2}} =\mathrm{1}.\mathrm{44}{kr}^{\mathrm{2}} \\ $$$$\:\%\:{Increase}\:{in}\:{A}=\frac{\mathrm{1}.\mathrm{44}{kr}^{\mathrm{2}} −{kr}^{\mathrm{2}} }{{kr}^{\mathrm{2}} }×\mathrm{100} \\ $$$$\:\:=\:\frac{.\mathrm{44}{kr}^{\mathrm{2}} }{{kr}^{\mathrm{2}} }×\mathrm{100}=\mathrm{44\%} \\ $$$$\:{V}\propto{r}^{\mathrm{3}} \Rightarrow{V}={Kr}^{\mathrm{3}} \:\left[{K}\:{variation}\:{constant}\right] \\ $$$$\:{V}_{{increase}} ={K}×\left(\mathrm{1}.\mathrm{2}{r}\right)^{\mathrm{3}} ={K}×\mathrm{1}.\mathrm{728}{r}^{\mathrm{3}} \\ $$$$\:\%\:{Increase}=\frac{\mathrm{1}.\mathrm{728}{Kr}^{\mathrm{3}} −{Kr}^{\mathrm{3}} }{{Kr}^{\mathrm{3}} }×\mathrm{100} \\ $$$$\:\:\:\:\:=\:\:\:\mathrm{72}.\mathrm{8\%} \\ $$$$ \\ $$
Commented by otchereabdullai@gmail.com last updated on 17/Jul/24
$${thank}\:{you}\:{sir}! \\ $$