BackA surface ship is moving (horizontally) in a straight line at 10 km/hr. At the same time, an enemy submarine maintains a position directly below the ship while diving at an angle that is 20° below the horizontal. How fast is the submarine’s altitude decreasing?
A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant. (Hint: Surface area=4πr².)
Area The area A of a triangle with sides of lengths a and b enclosing an angle of measure θ is
A = (1/2) ab sinθ.
a. How is dA/dt related to dθ/dt if a and b are constant?
If L = √(x² + y²), dx/dt = –1, and dy/dt = 3, find dL/dt when x = 5 and y = 12.
If y = x² and dx/dt = 3, then what is dy/dt when x = –1?
Right circular cone The lateral surface area S of a right circular cone is related to the base radius r and height h by the equation
______
S = πr √ r² + h².
c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant?
Consider the following cost functions.
c. Interpret the values obtained in part (b).
C(x) = 500+0.02x, 0≤x≤2000, a=1000
A jet flying at 450 mi/hr and traveling in a straight line at a constant elevation of 500 ft passes directly over a spectator at an air show. How quickly is the angle of elevation (between the ground and the line from the spectator to the jet) changing 2 seconds later?
Parabolic motion An arrow is shot into the air and moves along the parabolic path y=x(50−x) (see figure). The horizontal component of velocity is always 30 ft/s. What is the vertical component of velocity when (a) x=10 and (b) x=40? <IMAGE>
The volume V of a sphere of radius r changes over time t.
c. At what rate is the radius changing if the volume increases at 10 in³ when the radius is 5 inches?
Economics
Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x − 0.1x².
a. Find the average cost per machine of producing the first 100 washing machines.
If two opposite sides of a rectangle increase in length, how must the other two opposite sides change if the area of the rectangle is to remain constant?
Two boats leave a port at the same time, one traveling west at 20 mi/hr and the other traveling southwest ( 45° south of west) at 15 mi/hr. After 30 minutes, how far apart are the boats and at what rate is the distance between them changing? (Hint: Use the Law of Cosines.)
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = {x² − x, −2 ≤ x ≤−1
2x² − 3x − 3, −1 < x ≤ 0
Heating a plate When a circular plate of metal is heated in an oven, its radius increases at the rate of 0.01 cm/min. At what rate is the plate’s area increasing when the radius is 50 cm?