25. 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 ( frac{d V}{d x} ) when ( x=7 mathrm{~mm} ) and explain its meaning. ANSWER: ( quad frac{d V}{d x}(7)=147 frac{mathrm{mm}^{3}}{mathrm{~mm}} ) ( frac{d V}{d x} ) (7) represents the rate at which the volume is increasing with respect to the side length as ( x ) reaches ( 7 mathrm{~mm} ).
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Step 1: Recognize that the volume \( V \) of a cube with side length \( x \) is given by the formula \( V = x^3 \). 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 $d V / d x$ when $x=3 \mathrm{mm}$ 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 arguing by analogy with Exercise 53$(\mathrm{b})$ .
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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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In the shape of cubes, by allowing the solution of Sodium Chlorate crystals to grow, the volume of such a cube is chlorate and water to evaporate slowly. Suppose sodium 300 mm/hr. If the cube has a side length x, at what rate is the side decreasing at a rate of length decreasing at the instant x = 10 mm? 900 mm/hr 3 mm/hr mm/hr 10 mm/hr
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