A block of ice is exposed to heat in such a way that the block maintains a similar shape as it metls. The block of ice is initially 2' wide, 2 'high, and 3' long. If the change in the width of the ice is -1/3 ft/hr, find: the amount of time it will tak efor the block of the ice to completely melt.
I found that the rate of change in the volume of the ice block is -3/2 ft^3/hr. Do you just divide the volume of the cube (12 ft^3) by the rate of change of the volume (-3/2 ft^3/hr)?
Thank you! That was helpful :)
Well, first of all, I have to say that I'm impressed with your calculations. Who knew melting ice could be so mathematically satisfying?
But, uh-oh, we have a tiny problem here. It seems like you made a little mistake. You can't just divide the volume of the ice cube by the rate of change of the volume to find the time it will take to melt completely.
What you should do instead is divide the change in the volume of the ice block by the rate of change in the volume. Remember, the change in volume will be the final volume of the ice (which is zero, after it completely melts) minus the initial volume of the ice.
So, take that change in volume (-12 ft^3) and divide it by the rate of change in volume (-3/2 ft^3/hr). That will give you the amount of time it will take for the block of ice to completely melt.
And as a friendly reminder, please don't try to speed up the melting process by telling the ice some puns or cracking jokes at it. That might make things a little chilly.
Yes, you are correct. To find the amount of time it will take for the block of ice to completely melt, you can divide the initial volume of the ice block (12 ft^3) by the rate of change of the volume (-3/2 ft^3/hr):
Time = Volume / Rate of Change of Volume
Time = 12 ft^3 / (-3/2 ft^3/hr)
To divide a fraction, you can multiply by the reciprocal:
Time = 12 ft^3 * (-2/3 ft^3/hr)
The ft^3 units cancel out, leaving you with:
Time = 12 * (-2/3) hr
Time = -24/3 hr
Time = -8 hr
Therefore, it will take 8 hours for the block of ice to completely melt.
To find the amount of time it will take for the block of ice to completely melt, you can indeed use the volume and rate of change of volume. However, there is an error in the calculation of the rate of change of volume.
Let's go through the correct steps:
First, let's find the volume of the ice block. The initial dimensions given are 2ft wide, 2ft high, and 3ft long. So, the volume is calculated as:
Volume = width × height × length = 2ft × 2ft × 3ft = 12ft^3
Next, let's find the rate of change of the width of the ice block. It is given as -1/3 ft/hr, which means the width is decreasing by 1/3 ft every hour.
Now, let's calculate the rate of change of volume. Since the width is changing, the volume will change as well. We can express the volume as a function of time, where V(t) represents the volume of the ice block at time t. In this case, V(t) = width(t) × height × length.
The width, at time t, will be the initial width (2ft) minus the change in width (-1/3 ft/hr) multiplied by time (t):
width(t) = 2ft - (1/3ft/hr) × t
Now, substitute this expression for width(t) back into the volume equation:
V(t) = (2ft - (1/3ft/hr) × t) × 2ft × 3ft
Simplifying the expression, we get:
V(t) = (12ft^2 - (2/3ft^2/hr) × t) × 3ft
Differentiating V(t) with respect to t to find the derivative of V(t), we get:
dV(t)/dt = -(2/3ft^2/hr) × 3ft
The negative sign indicates that the volume is decreasing over time.
Now, substitute V(t) = 12ft^3 and dV(t)/dt = -(2/3ft^2/hr) × 3ft into the differential equation:
-(2/3ft^2/hr) × 3ft = dV(t)/dt = 12ft^3
Simplifying further:
-2ft/hr = 12ft^3
Now, divide both sides by -2ft/hr to isolate t:
t = 12ft^3 / -2ft/hr
Simplifying,
t = -6hr/ft
The negative sign indicates that time cannot be negative, so we disregard it:
t = 6hr/ft
Therefore, it will take 6 hours per foot for the block of ice to completely melt.
v = w*w*3w/2 = 3/2 w^3
dv/dt = 9/2 w^2 dw/dt
= 9/2 * -1/3 = -3/2
your dv/dt is correct.
as for your question, consider the units:
time = ft^3 / ft^3/hr = 12/(3/2) = 8 hr