Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3a

Chapter 3, Problem 3a

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?

Verified step by step guidance

Start by recalling the formula for the area of a rectangle, which is given by \( A = l \times w \), where \( l \) is the length and \( w \) is the width.

Since the area \( A \) is to remain constant, we can express this as \( l_1 \times w_1 = l_2 \times w_2 \), where \( l_1 \) and \( w_1 \) are the original dimensions, and \( l_2 \) and \( w_2 \) are the new dimensions.

If two opposite sides (say the lengths) increase, then \( l_2 > l_1 \). To keep the area constant, the product \( l_2 \times w_2 \) must equal \( l_1 \times w_1 \).

To maintain the equality, the width \( w_2 \) must decrease such that \( w_2 = \frac{l_1 \times w_1}{l_2} \). This ensures that the product of the new dimensions equals the original area.

Thus, as the length increases, the width must decrease proportionally to maintain the constant area of the rectangle.

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Key Concepts

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

Area of a Rectangle

The area of a rectangle is calculated by multiplying its length by its width (A = l × w). This fundamental formula is crucial for understanding how changes in the dimensions of the rectangle affect its overall area. If one dimension increases, the other must adjust to maintain a constant area.

Inverse Relationship

An inverse relationship occurs when one quantity increases while another decreases, such that their product remains constant. In the context of the rectangle, if the lengths of two opposite sides increase, the lengths of the other two sides must decrease proportionally to keep the area unchanged.

Proportionality

Proportionality refers to the relationship between two quantities where a change in one quantity results in a corresponding change in another. In this scenario, the lengths of the opposite sides of the rectangle are proportional to each other, meaning that if one pair of sides increases, the other pair must decrease in a specific ratio to maintain the constant area.

Related Practice

Textbook Question

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

³√x+³√y⁴ = 2;(1,1)

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Textbook Question

Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure). <IMAGE>

a. Express the angle of elevation θ from the biologist to the falcon as a function of the height h of the bird above the ground. (Hint: The vertical distance between the top of the cliff and the falcon is 80−h.)

Textbook Question

Airline travel The following figure shows the position function of an airliner on an out-and-back trip from Seattle to Minneapolis, where s = f(t) is the number of ground miles from Seattle t hours after take-off at 6:00 A.M. The plane returns to Seattle 8.5 hours later at 2:30 P.M. <IMAGE>

a. Calculate the average velocity of the airliner during the first 1.5 hours of the trip (0 ≤ t ≤ 1.5).

Textbook Question

21–30. Derivatives

a. Use limits to find the derivative function f' for the following functions f.

f(x) = 4x²+1; a= 2,4

Textbook Question

62–65. {Use of Tech} Graphing f and f'

a. Graph f with a graphing utility.

f(x)=e^−x tan^−1 x on [0,∞)

Textbook Question

62–65. {Use of Tech} Graphing f and f'

a. Graph f with a graphing utility.

f(x) = (x−1) sin^−1 x on [−1,1]