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05:14Resistors 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?
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.
a. How is dS/dt related to dr/dt if h is constant?
Temperature and the period of a pendulum For oscillations of small amplitude (short swings), we may safely model the relationship between the period T and the length L of a simple pendulum with the equation:
T = 2π√(L/g),
where g is the constant acceleration of gravity at the pendulum’s location. If we measure g in centimeters per second squared, we measure L in centimeters and T in seconds. If the pendulum is made of metal, its length will vary with temperature, either increasing or decreasing at a rate that is roughly proportional to L. In symbols, with u being temperature and k the proportionality constant,
dL/du = kL.
Assuming this to be the case, show that the rate at which the period changes with respect to temperature is kT/2.
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
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S = πr √ r² + h².
c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant?
If x¹/³ + y¹/³ = 4, find d²y/dx² at the point (8, 8).
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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