A perfect cube shaped ice cube melts so that the length of its sides are decreasing at a rate of 2in//sec. Assume that the block retains its cube shape as it melts. At what rate is the volume of the ice cube changing when the sides are 7 in each?

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A perfect cube shaped ice cube melts so that the length of its sides are decreasing at a rate of  2in//sec. Assume that the block retains its cube shape as it melts. At what rate is the volume of the ice cube changing when the sides are 7 in each?

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Volume Formula: Let's denote the side length of the cube as s (in inches). The volume V of a cube is given by the formula V = s^3. We are given that the side length s is decreasing at a rate of 2 inches per second, which we can write as ds/dt = -2 in/sec (the negative sign indicates a decrease). We want to find the rate of change of the volume, dV/dt, when s = 7 inches.Chain Rule Application: To find dV/dt, we will use the chain rule from calculus, which allows us to relate the rates of change of different quantities. The chain rule tells us that dV/dt = (dV/ds) * (ds/dt). We already know ds/dt, so we need to find dV/ds first.Finding dV/ds: Differentiating the volume formula V = s^3 with respect to s, we get dV/ds = 3s^2. This derivative represents the rate of change of the volume with respect to the side length.Substitute s=7: Now we can substitute s = 7 inches into the derivative to find the rate of change of the volume when the side length is 7 inches. So, dV/ds = 3 * (7 in)^2 = 3 * 49 in^2 = 147 in^2.Calculate dV/dt: We can now use the chain rule to find dV/dt. We have dV/dt = (dV/ds) * (ds/dt) = 147 in^2/sec * (-2 in/sec) = -294 in^3/sec. The negative sign indicates that the volume is decreasing.Final Result: The rate at which the volume of the ice cube is changing when the sides are 7 inches each is -294 cubic inches per second.

A perfect cube shaped ice cube melts so that the length of its sides are decreasing at a rate of $2\frac{\text{in}}{\text{sec}}$ . Assume that the block retains its cube shape as it melts. At what rate is the volume of the ice cube changing when the sides are $7\text{in}$ each?

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A perfect cube shaped ice cube melts so that the length of its sides are decreasing at a rate of 2insec2\frac{\text{in}}{\text{sec}}2secin​. Assume that the block retains its cube shape as it melts. At what rate is the volume of the ice cube changing when the sides are 7in7\text{in}7in each?

A perfect cube shaped ice cube melts so that the length of its sides are decreasing at a rate of $2\frac{\text{in}}{\text{sec}}$ . Assume that the block retains its cube shape as it melts. At what rate is the volume of the ice cube changing when the sides are $7\text{in}$ each?