Question

The surface area of a balloon bearing slogan STOP CHILD LABOUR being inflated changes at a constant rate. If initially, its radius is 3 units and after 2 seconds, it is 5 units, find the radius after ‘t’ seconds. Why child labour is not good for society? \[({r^2} = 8t + 9)\]

Answer 1

Hint: Firstly we have to find the derivative of surface area, and then integrate it. After it put the value of t and r in the integrated value of surface area then we get the value of C. Then put the value of C in the integrated value of surface area, we get the value of K. Secondly, put the value of t and r in the equation in (III) to get the value of C. After getting the value of C, Put the value of C in Equation (III) to get the value of K. Put the value of K and C in equation (a) to get the value of r. Then put the value of K and C in equation (b) to get the value of r.

Complete step-by-step answer:
Here we are given that the surface area of a balloon bearing slogan STOP CHILD LABOUR being inflated changes at a constant rate.
It is given that the initial radius is 3 units and after 2 seconds, it is 5 units.
So, we have to find the radius after t seconds.
Surface area, A = $4\pi {r^2}$
$\dfrac{{dA}}{{dr}} = kt$ (Given)
Here, A=$4\pi {r^2}$
$\dfrac{{dA}}{{dr}} = 4\pi \times 2r \cdot \dfrac{{dr}}{{dt}}$
Now, on integrate the equation on both the sides
$\int {kt \cdot dt = \int {8\pi r \cdot dr} } $
$\Rightarrow$ $k\int {t \cdot dt = 8\pi \int {r.dr} } $
$\Rightarrow$ $\dfrac{{8\pi {r^2}}}{2} = k{t^2} + C$ _(I)
At t=0 , r=3
We have to put the value of t and r in the equation (I)
$\Rightarrow$ $\dfrac{{8\pi \times 9}}{2} = 0 + C$
$\Rightarrow$ $\therefore $C=$36\pi $
Now, put the value of C in equation(II)
So, $4\pi {r^2} = k{t^2} + 36\pi $_(II)
At t=0, r=5
Now put the value of t and r in the equation (II)
$4\pi \times 25 = k \times 4 + 36\pi $
$\Rightarrow$ $100\pi = 4k + 36\pi $
$\Rightarrow$ $k = 16\pi $
Here, \[\dfrac{{8\pi {r^2}}}{2} = \dfrac{{k{t^2}}}{2} + C\]
$\Rightarrow$ \[\dfrac{{4\pi {r^2}}}{2} = \dfrac{{k{t^2}}}{2} + 2C\]
$\Rightarrow$ $8\pi {r^2} = k{t^2} + C$ _(III)
At t=0, r=3
Put the value of t and r in equation (III)
$\Rightarrow$ $8\pi \times 9 = 0 + C$
$\Rightarrow$ $72\pi = 0 + C$
$\Rightarrow$ $C = 72\pi $
Now, put the value of C in equation (III)
$\Rightarrow$ $8\pi {r^2} = k{t^2} + 72\pi $ _(IV)
At t=2, r=5
Put the value of t and r in the equation(IV)
$\Rightarrow$ $8\pi \times 25 = 4k + 72\pi $
$\Rightarrow$ $200\pi = 4k + 72\pi $
$\Rightarrow$ $128\pi = 4k$
$\Rightarrow$ $k = 64\pi $
 From the equation
$4\pi {r^2} = k{t^2} + C$ _(a)
Now, put the value of k = 16 and C = 36 in the above equation
$\Rightarrow$ $4\pi {r^2} = 16\pi {t^2} + 36C$
$\Rightarrow$ ${r^2} = 4{t^2} + 9$
$\Rightarrow$ $r = \sqrt {4{t^2} + 9} $
From the equation
$8\pi {r^2} = k{t^2} + C$ _(b)
Now, put the value of k = $64\pi $ and C = $72\pi $ in the above equation
$\Rightarrow$ $8\pi {r^2} = 64\pi {t^2} + 72\pi $
$\Rightarrow$ ${r^2} = 8{t^2} + 9$
$\Rightarrow$ $r = \sqrt {8{t^2} + 9} $

Note: Child labour is not good for society because without an education, children grow up without the skills they need to secure employment, making it more likely that they'll send their own children to work someday. This cycle must end. Stopping child labour creates a better world for children and adults.