Find r as a function of time t.
r starts at 38 inches and steadily increases at 0.16 inch per second. So in t seconds r would have increased by 0.16t when added to the initial 38 inches will give:
r(t) = 38 + 0.16t
Volume of a sphere = 4/3(pi)r^3. Since radius changes with time, volume will change with time too.
V(t) = 4/3(pi)r^3 = 4/3(pi)(38 + 0.16t)^3
Put t=280 and solve to get V
V=2377823.53
You can use the fact that the volume of the balloon requires the radius and that we have the rate of the increment of the radius by which the actual function of r in terms of t can be retrieved.
Suppose the rate of a function is given as some constant,
and let the function be f(x), then we have:
[tex]\dfrac{d(f(x))}{dx} = k[/tex], where k is a constant, then we have:
[tex]\text{Integrating both the sides with respect to x, we get}\\\\\int \dfrac{d(f(x))}{dx}dx = \int kdx\\\\f(x) = x+ c[/tex]
where c is constant of integration.
Since the radius of the balloon is function of time, thus we have:
[tex]r = f(t)[/tex]
Since the rate of increment of radius with respect to time is 0.16 inches per second, thus,
[tex]\dfrac{dr}{dt} = 0.16[/tex]
Integrating both the sides with respect to 't', we get:
[tex]\int \dfrac{dr}{dt} dt = \int 0.16 dt\\\\r = 0.16t + c[/tex]
Since it was given that at time t = 0, we had radius of the spherical balloon as 38 inches, thus, putting it in the obtained equation, we get:
[tex]r = 0.16t + c\\38 = 0.16 \times 0 + c\\38 = c[/tex]
Thus, the constant of integration here is 38
Thus, the radius of the balloon at time t is given by the equation
[tex]r = 0.16t + 38[/tex]
Now, since the volume of sphere with radius r units is given by [tex]\dfrac{4}{3} \pi r^3[/tex] cubic units, thus, we have;
[tex]V_{balloon} = \dfrac{4}{3} \times \pi \times r = \dfrac{4}{3}\pi(0.16t + 38)^3 \\\\V_{balloon} = \dfrac{4}{3}\pi(0.16t + 38)^3[/tex]
For getting volume at time t = 280 seconds, we use above equation, thus,
[tex]V = \dfrac{4}{3}\pi(0.16t + 38)^3 \\\\V|_{t = 280} =\dfrac{4}{3}\pi(0.16 \times 280 + 38)^3 \approx 2377823.526[/tex]
Thus,
Learn more about rate of change here:
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