Just sub the value of r into the volume formula
V = (4/3)Ï€(50 - 2.5t)^3
expand if necessary, I wouldn't.
V = (4/3)Ï€(50 - 2.5t)^3
expand if necessary, I wouldn't.
V(t) = (4/3) * pi * (50 - 2.5t)^3 cm³
Now that's a snowball with potential!
Given that r = 50 - 2.5t, we can substitute this into the volume formula V = (4/3)(Ï€)(r^3) to get:
V = (4/3)(Ï€)(50 - 2.5t)^3
Simplifying further:
V = (4/3)(Ï€)(125000 - 3750t + 37.5t^2 - 0.125t^3)
Thus, the formula for V as a function of time is:
V = 166666.67Ï€ - 12500Ï€t + 125Ï€t^2 - (4/3)Ï€t^3
Given that the radius, r, is given by r = 50 - 2.5t, we can substitute this expression into the formula for the volume:
V = (4/3) * π * r^3
Replacing r with the expression 50 - 2.5t:
V = (4/3) * π * (50 - 2.5t)^3
Now, we can simplify the expression to get the formula for V as a function of time. Let's start by expanding the cube:
V = (4/3) * π * (50 - 2.5t) * (50 - 2.5t) * (50 - 2.5t)
V = (4/3) * π * (50 - 2.5t) * (2500 - 250t + 6.25t^2)
Next, we can distribute and simplify further:
V = (4/3) * π * (125000 - 12500t + 312.5t^2 - 125t + 12.5t^2 - 0.3125t^3)
V = (4/3) * π * (125000 - 13750t + 324.5t^2 - 0.3125t^3)
Finally, we can arrange the terms in descending order of powers of t and express it as a function of time:
V = (4/3) * π * (-0.3125t^3 + 324.5t^2 - 13750t + 125000)
Therefore, the formula for V as a function of time is:
V(t) = (4/3) * π * (-0.3125t^3 + 324.5t^2 - 13750t + 125000)