Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate the derivative when x = 5 mm. V'(5) = [ ] mm^3/mm What does V'(5) mean in this situation? V'(5) represents the rate at which the side length is increasing with respect to the volume as x reaches 5 mm. V'(5) represents the rate at which the volume is increasing as x reaches 15 mm. V'(5) represents the rate at which the volume is increasing with respect to the side length as x reaches 5 mm. V'(5) represents the volume as the side length reaches 5 mm. V'(5) represents the rate at which the volume is increasing with respect to the side length as V reaches 15 mm^3.
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First, we know that the volume of a cube is given by the formula: $V = x^3$ Now, we want to find the derivative of V with respect to x: $\frac{dV}{dx} = \frac{d}{dx}(x^3) = 3x^2$ Show more…
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(a) Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If $ V $ is the volume of such a cube with side length $ x. $ calculate $ dV/dx $ when $ x = 3 $ mm and explain its meaning. (b) Show that the rate of change of the volume of a cube with respect to its edge length is equal to half the surface area of the cube. Explain geometrically why this result is true by analogy with Exercise 11(b).
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Rates of Change in the Natural and Social Sciences
Sodium Chlorate crystals are easy to grow in the shape of cubes by allowing a solution of sodium chlorate and water to evaporate slowly. Suppose that the volume of such a cube is decreasing at a rate of 300 mm³/hr. If the cube has a side length x, at what rate is the side length decreasing at the instant x = 10 mm? 10 mm/hr 1 mm/hr 900 mm/hr 3 mm/hr
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