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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 103a
Chapter 3, Problem 103a

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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Verified step by step guidance
1
Identify the geometric shape of the tank: The tank is conical, which means it has a circular base and tapers to a point.
Understand the relationship between the height (h) and the radius (r) of the cone: In similar triangles, the ratio of corresponding sides is constant.
Use the properties of similar triangles: If the full height of the cone is H and the full radius is R, then for any height h, the radius r can be expressed as r = (R/H) * h.
Express the relationship between h and r using the formula for similar triangles: r = (R/H) * h, where R and H are constants representing the dimensions of the full cone.
Ensure the units are consistent: Since the rate of water drainage is given in cubic feet per minute, make sure h and r are in feet to maintain consistency in calculations.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Volume of a Cone

The volume of a cone is given by the formula V = (1/3)πr²h, where r is the radius of the base and h is the height. Understanding this formula is crucial for relating the volume of water in the tank to the height and radius as the tank drains.
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Finding Volume Using Disks

Related Rates

Related rates involve finding the rate at which one quantity changes in relation to another. In this problem, as water drains from the tank, the height (h) and radius (r) of the water's surface change, and we need to establish a relationship between these rates to solve the problem.
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Intro To Related Rates

Geometric Relationships

In a conical tank, the relationship between the height and radius of the water level can often be expressed as a proportionality based on similar triangles. This geometric relationship allows us to express r in terms of h, which is essential for applying the volume formula and finding the rate of change.
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Derivatives Applied To Velocity
Related Practice
Textbook Question

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?

Textbook Question

Find the linearizations of


a. tan x at x = -π/4


Graph the curves and linearizations together.

Textbook Question

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?

Textbook Question

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). 

a. How is dS/dt related to dr/dt if h is constant?

Textbook Question

Find the linearization of ƒ(x) = √(1 + x) + sin x - 0.5 at x = 0.

Textbook Question

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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