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.

A company makes computer chips from square wafers of silicon. It wants to keep the side length of a wafer very close to 15 mm and it wants to know how the area $A(x)$ of a wafer changes when the side length $x$ changes. Find $A^{\prime}(15)$ and explain its meaning in this situation.

The height (in meters) of a projectile shot vertically upward from a point 2m above ground level with an initial velocity of 24.5 m/s is h = 2 + 24.5t - 4.9t 2 after t seconds. (a) Find the velocity after 2s and after 4s. (b) When does the projectile reach its maximum height? (c) What is the maximum height? (d) When does it hit the ground? (e) With what velocity does it hit the ground?

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s. Find the rate at which the area within the circle is increasing after 1 s.

A particle moves according to a law of motion s=f(t), t≥0, where t is measured in seconds and s in feet. Find the acceleration at time t and after 3 s.

Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing two hours later?

The height (in meters) of a projectile shot vertically upward from a point 2 m above ground level with an initial velocity of 24.5 m/s is h = 2 + 24.5t - 4.9t^2 after seconds. When does the projectile reach its maximum height?

The height (in meters) of a projectile shot vertically upward from a point 2 m above ground level with an initial velocity of 24.5 m/s is h = 2 + 24.5t - 4.9t^2 after seconds. What is the maximum height?

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.

A cylindrical tank with radius 7 m is being filled with water at a rate of 2 m3/min. How fast is the height of the water increasing?

A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?.

A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?

Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes. x^2+y^2=r^2, ax+by=0

Minimum Length The wall of a building is to be braced by a beam that must pass over a parallel fence $5$ feet high and $4$ feet from the building. Find the length of the shortest beam that can be used.