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

{Use of Tech} A damped oscillator The displacement of a mass on a spring suspended from the ceiling is given by $y = 10 e^{- \frac{t}{2}} \cos (\frac{π t}{8})$ .

a. Graph the displacement function.

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Textbook Question

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = 8 - 2x²; P(0, 8)

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Textbook Question

7–14. Find the derivative the following ways:

a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.

g(s) = 4s³ - 8s² +4s / 4s

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Textbook Question

31–32. Velocity functions A projectile is fired vertically upward into the air, and its position (in feet) above the ground after t seconds is given by the function s(t).

a. For the following functions s(t), find the instantaneous velocity function v(t). (Recall that the velocity function v is the derivative of the position function s.)

s(t)= −16t²+100t

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Textbook Question

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

(x+y)^2/3=y; (4, 4)

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Textbook Question

Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

f(x) = 2/√x; P(4,1)

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