how long will it take an ice cube to melt completely? You left out an ice cube ant 1/4 of it melts in 1 hour. Use the following steps to find out how long it will take before it is melted completely.
1-let s represent the side length of the ice cube, and observe that s is a function of time t. Write expressions which describe the volume V and surface area A of the cube as functions of s (which you should observe, in turn makes them functions of t).
2-melting takes place at the surface of the cube, so you decided that it is reasonable to assume that the cubes volumn V decreases at a rate that is proportional to its surface area A. saying that a is proportional to b means that there is constant k so that a=kb.) write an equation that describes dV/dt in terms of s. Is your constant k a positive or negative number?
3-use the chain rule and your answer from #1 to write an expression which relates dV/dt to ds/dt
4-now you have two expressions (your answers from 2 and 3 for dV/dt).set them equal to find ds/dt in terms of the constant k.
5-use your answer from #4 to write an equation which relates s0=s(0) to s1=s(1), where t is measured in hours, and use it to find tmelt (the melting time) in terms of the quantity s1/s0.
6-taking into account that 1/4 of the cube melted in 1 hour, find an approximation to s1/s0 using the function you wrote in #1 for V in terms of s.
7-find tmelt. how much longer will you have to wait for the ice cube to melt completely?
To solve this problem, let's follow the steps provided:
1. Let s represent the side length of the ice cube, and observe that s is a function of time t. The volume V of a cube is given by V = s^3, and the surface area A is given by A = 6s^2. Therefore, both V and A can be expressed as functions of s and, in turn, as functions of t.
2. Since melting takes place at the surface of the cube, we assume that the volume V decreases at a rate that is proportional to its surface area A. This can be expressed as follows: dV/dt = kA. Here, k is a constant. To determine if k is positive or negative, consider that as time increases, the ice cube is melting. This means the volume is decreasing, so dV/dt should be negative. Since A is always positive, the constant k must be negative.
3. Now, let's use the chain rule to relate dV/dt to ds/dt. Since V = s^3, we can differentiate both sides with respect to time (t): dV/dt = d(s^3)/dt. By the chain rule, we get dV/dt = 3s^2 * ds/dt.
4. Setting dV/dt from step 2 equal to dV/dt from step 3, we have kA = 3s^2 * ds/dt. This equation relates ds/dt to the constant k.
5. To find tmelt (the melting time) in terms of the quantity s1/s0, we need the equation relating s0 to s1. Since 1/4 of the cube melts in 1 hour, we know that s1 = (3/4)s0. Let's rearrange this equation to solve for s0: s0 = (4/3)s1.
6. Using the equation V = s^3 from step 1, we can express V1 (volume at t = 1 hour) and V0 (initial volume) as functions of s1 and s0, respectively. V1 = (s1)^3 and V0 = (s0)^3. Let's find s1/s0.
s1/s0 = (s1)^3 / (s0)^3
= ((3/4)s0)^3 / (s0)^3
= (3/4)^3
= 27/64
7. Now, let's find the melting time tmelt using the equation derived in step 5. Using s1/s0 = 27/64, we have:
s0 = (4/3)s1
s0 = (4/3)(27/64)s0
1 = (4/3)(27/64)t_melt
Solving for t_melt, we get:
t_melt = 1 / [(4/3)(27/64)]
= 64 / (4/3)(27)
= 64 / (108/12)
= 64 * 12 / 108
= 32/3
≈ 10.67 hours
Therefore, it will take approximately 10.67 hours for the ice cube to melt completely.