what is the volume of an ice cube with an initial length of 5cm after 4 minutes when the ice cube is melting at a rate of 0.6x^2cm^3 per minute?

Answer 1

Okay. Here is the approach to use :

1. First put down what you have : x= 5cm, t=4 min, dv/dt (Rate at which the volume of the cube changes by melting)=- 0.6x^2 cm^3/min. Note that the rate of change of the cube's volume is a function on it's own and also note that it is negative. The negative sign means that its volume is decreasing, implying that it is melting.
2. So find dv/dt at x=5, dv/dt (at x = 5) =- 0.6(5)^2= -15 cm^3/min. This means that when the length of the ice cube is 5 cm, the volume of the ice cube would be decreasing by -15 cm^3/min
3. Next, use the formula for the Volume of the cube. V = x^3
4. Now find the derivative of each of them with respect to time, treating V and x as independent functions.
5. So we have dV/dt = 3x^2 (dx/dt) .
6. Now plug in the values you have to get -15 = 3(5)^2 (dx/dt) . To result in -15 = 75 * (dx/dt).
7. So finally we get : dx/dt = -15/75 = -1/5 cm/min = -0.2 cm/min.
8. Now find the change in length after 4 mins.(Hint : use dt = 4) . So we have dx/4 = -0.2 , Δx = -0.8 cm
9. Now you will have to go to this formula dV/dx = ΔV/Δx.
10. Find dV/dx. dV/dx = 3x^2. At x=5 cm, dV/dx = 3(5)^2 = 75 cm.
11. Put dV/dx back into step 9, with Δx = -0.8 cm. So this would result in 75 = ΔV/(-0.8).
12. Solve for ΔV, ΔV=(75)(-0.8) = -60 cm^3.
13. Whew! Final Step : Final volume = Initial Volume + Change in Volume.This is V_f = V_i + ΔV = 5^3 - 60 = 25 -60 = -35 cm^3
14. The negative Volume only means that by the 4 minutes all of the Ice cube would have melted. So after 4 minutes we expect to have 0 cm^3 of volume left. So practically the answer by reasoning is 0 cm^3 because all of the ice would have melted by that time.
15. I hope this helps!