An ice cube is melting such that the side of the cube is decreasing at the rate of 1/4 t^2 cm / min. It is given that the side of the cube is 8cm at the start of the experiment. Calculate the length of the side of the cube when t = 4? ( answer 2 2/3 cm)
ds/dt = -(1/4) t^2
s = -(1/12) t^3 + 8
when t = 4
s = - (1/12)(4*16) + 8
= -16/3 + 24/3
= 8/3
= 2 2/3
To solve this problem, we can use calculus and the given rate of change of the side of the cube to find the length of the side when t = 4.
Let's start by setting up the problem:
Given:
Rate of change of the side of the cube: ds/dt = -1/4t^2 cm/min
Initial side length: s(0) = 8 cm
Time when we want to find the side length: t = 4
We need to find the side length s(4) when t = 4.
To find s(4), we can integrate the rate of change equation from t = 0 to t = 4:
∫ (ds/dt) dt = ∫ (-1/4t^2) dt
Integrating both sides gives:
∫ ds = ∫ (-1/4t^2) dt
Integrating -1/4t^2 with respect to t gives:
s = 1/4 ∫ (1/t^2) dt
s = 1/4 * (-1/t) + C
where C is the constant of integration.
Now, let's solve for C using the initial condition s(0) = 8:
8 = 1/4 * (-1/0) + C
8 = 0 + C
C = 8
Substituting C back into the equation, we have:
s = 1/4 * (-1/t) + 8
Now we can find s(4) by plugging in the value of t = 4:
s(4) = 1/4 * (-1/4) + 8
s(4) = -1/16 + 8
s(4) = -1/16 + (8*16)/16
s(4) = -1/16 + 128/16
s(4) = (128 - 1) / 16
s(4) = 127/16
Therefore, when t = 4, the length of the side of the cube is 127/16 cm, which can be simplified to 7 15/16 cm or approximately 7.94 cm.