# The spherical balloon is inflated at the rate of 10 m³/sec. find the rate at which the surface area is increasing when the radius of the sphere is 3m? The correct answer was given: renee9913

the balloon has a volume $$v$$ dependent on its radius $$r$$:

$$v(r)=\dfrac43\pi r^3$$

differentiating with respect to time $$t$$ gives

$$\dfrac{\mathrm dv}{\mathrm dt}=4\pi r^2\dfrac{\mathrm dr}{\mathrm dt}$$

if the volume is increasing at a rate of 10 cubic m/s, then at the moment the radius is 3 m, it is increasing at a rate of

$$10\dfrac{\mathrm m^3}{\mathrm s}=4\pi (3\,\mathrm m)^2\dfrac{\mathrm dr}{\mathrm dt}\implies\dfrac{\mathrm dr}{\mathrm dt}=\dfrac5{18\pi}\dfrac{\rm m}{\rm s}$$

the surface area of the balloon is

$$s(r)=4\pi r^2$$

and differentiating gives

$$\dfrac{\mathrm ds}{\mathrm dt}=8\pi r\dfrac{\mathrm dr}{\mathrm dt}$$

so that at the moment the radius is 3 m, its area is increasing at a rate of

$$\dfrac{\mathrm ds}{\mathrm dt}=8\pi(3\,\mathrm m)\left(\dfrac5{18\pi}\dfrac{\rm m}{\rm s}\right)=\dfrac{20}3\dfrac{\mathrm m^2}{\rm s}$$ The correct answer was given: Brain
The answer would be A. 1 1/5

48/40 = 1 8/40

1 8/40 Simplify to 1 1/5

Hope this helps! The correct answer was given: Brain
I believe it is 60-10=x because it says she purchased it with 60$but recieved 10$ in change causing the amount we dont know to be a variable such as “x”