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?
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?
Resistors connected in parallel If two resistors of R₁ and R₂ ohms are connected in parallel in an electric circuit to make an R-ohm resistor, the value of R can be found from the equation
1/R = 1/R₁ + 1/R₂
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If R₁ is decreasing at the rate of 1ohm/sec and R₂ is increasing at the rate of 0.5 ohm/sec, at what rate is R changing when R₁ = 75 ohms and R₂ = 50 ohms?
Draining a tank Water drains from the conical tank shown in the accompanying figure at the rate of 5 ft³/min.
a. What is the relation between the variables h and r in the figure?
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Moving searchlight beam The figure shows a boat 1 km offshore, sweeping the shore with a searchlight. The light turns at a constant rate, dθ/dt = -0.6 rad/sec.
b. How many revolutions per minute is 0.6 rad/sec?
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Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.
b. How is dS/dt related to dh/dt if r is constant?
Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.
c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant?
Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.
d. How is dr/dt related to dh/dt if S is constant?
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.
b. How is dS/dt related to dh/dt if r is constant?
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.
Economics
Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x − 0.1x².
c. Show that the marginal cost when 100 washing machines are produced is approximately the cost of producing one more washing machine after the first 100 have been made, by calculating the latter cost directly.
Economics
Marginal revenue
Suppose that the revenue from selling x washing machines is
r(x) = 20000(1 − 1/x) dollars.
b. Use the function r'(x) to estimate the increase in revenue that will result from increasing production from 100 machines a week to 101 machines a week.
Economics
Marginal revenue
Suppose that the revenue from selling x washing machines is
r(x) = 20000(1 − 1/x) dollars.
c. Find the limit of r'(x) as x → ∞. How would you interpret this number?
If y = x² and dx/dt = 3, then what is dy/dt when x = –1?
If L = √(x² + y²), dx/dt = –1, and dy/dt = 3, find dL/dt when x = 5 and y = 12.
If r + s² + v³ = 12, dr/dt = 4, and ds/dt = –3, find dv/dt when r = 3 and s = 1.