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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.43b
Chapter 3, Problem 3.43b

City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.
b. How fast will the city be growing when it reaches a size of 38 mi²?

Verified step by step guidance
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Step 1: Identify the function A(t) = -\(\frac{1}{50}\)t^2 + 2t + 20, which models the size of the city in square miles as a function of time t in years after 2010.
Step 2: To find how fast the city is growing, we need to determine the rate of change of A with respect to t, which is the derivative A'(t).
Step 3: Differentiate A(t) with respect to t to find A'(t). The derivative of A(t) = -\(\frac{1}{50}\)t^2 + 2t + 20 is A'(t) = -\(\frac{2}{50}\)t + 2.
Step 4: Set A(t) equal to 38 to find the value of t when the city reaches a size of 38 mi². Solve the equation -\(\frac{1}{50}\)t^2 + 2t + 20 = 38 for t.
Step 5: Once you have the value of t, substitute it into A'(t) to find the rate of growth of the city at that time.

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Key Concepts

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

Function Analysis

Understanding the function A(t) = -1/50t² + 2t + 20 is crucial for analyzing the growth of the city. This quadratic function represents the area of the city over time, where the coefficients indicate how the area changes as time progresses. The vertex of the parabola can provide insights into maximum growth, while the shape of the graph helps in understanding the growth rate at different time intervals.
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Derivatives Applied To Velocity

Derivative and Rate of Change

The derivative of the function A(t) represents the rate of change of the city's area with respect to time. By calculating A'(t), we can determine how fast the city is growing at any given time. This concept is essential for finding the growth rate when the city reaches a specific size, such as 38 mi², as it allows us to evaluate the instantaneous growth at that point.
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Solving for Specific Values

To find out how fast the city is growing when it reaches 38 mi², we first need to solve the equation A(t) = 38 for t. This involves setting the function equal to 38 and solving for the time variable t. Once we have the appropriate value of t, we can substitute it back into the derivative A'(t) to find the growth rate at that specific moment.
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Related Practice
Textbook Question

Owlet talons Let L (t) equal the average length (in mm) of the middle talon on an Indian spotted owlet that is t weeks old, as shown in the figure.<IMAGE>

b. Estimate the value of L'(a) for a ≥ 4 . What does this tell you about the talon lengths on these birds? (Source: ZooKeys, 132, 2011)

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Textbook Question

Volume of a torus The volume of a torus (doughnut or bagel) with an inner radius of a and an outer radius of b is V=π²(b+a)(b−a)²/4.

b. Evaluate this derivative when a=6 and b=10.

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Textbook Question

{Use of Tech} A mixing tank A 500-liter (L) tank is filled with pure water. At time t=0, a salt solution begins flowing into the tank at a rate of 5 L/min. At the same time, the (fully mixed) solution flows out of the tank at a rate of 5.5 L/min. The mass of salt in grams in the tank at any time t≥0 is given by M(t) = 250(1000−t)(1−10−³⁰(1000−t)¹⁰) and the volume of solution in the tank is given by V(t) = 500-0.5t.

b. Graph the volume function and verify that the tank is empty when t=1000 min. 

Textbook Question

The Chain Rule for second derivatives

b. Use the formula in part (a) to calculate d2dx2(sin(3x4+5x2+2))\(\frac{d^2}{dx^2}\[\left\)(\(\sin\]\left\)(3x^4+5x^2+2\(\right\))\(\right\)).

Textbook Question

Deriving trigonometric identities

b. Verify that you obtain the same identity for sin2t as in part (a) if you differentiate the identity cos 2t = 2 cos² t−1.

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Textbook Question

58–59. Carry out the following steps.

b. Find the slope of the curve at the given point.

xy^5/2+x^3/2y=12; (4, 1)