Textbook Question
Suppose S = x + 2y is an objective function subject to the constraint xy = 50, for x > 0 and y > 0.
b. Find the absolute minimum value of S subject to the given constraint.
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 4.5.16.b
Pen problems
b. A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of 100 m² (see figure). What are the dimensions of each pen that minimize the amount of fence that must be used? <IMAGE>
Verified step by step guidance1
Define the variables: Let the width of each pen be 'w' meters and the length be 'l' meters. Since there are four pens, the total length along the barn is '4w'.
Set up the area equation: Each pen has an area of 100 m², so the equation for the area of one pen is 'w * l = 100'.
Express 'l' in terms of 'w': From the area equation, solve for 'l' to get 'l = 100 / w'.
Set up the perimeter equation: The total amount of fencing needed is the sum of the lengths of the sides not against the barn. This is '3l + 5w' (3 lengths and 5 widths).
Substitute 'l' in the perimeter equation: Replace 'l' with '100 / w' in the perimeter equation to get '3(100 / w) + 5w'. Simplify this expression to find the function of 'w' that represents the total fencing needed. Then, find the derivative and solve for 'w' to minimize the fencing.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Optimization
Optimization in calculus involves finding the maximum or minimum values of a function. In this problem, the rancher seeks to minimize the total length of the fence while maintaining a fixed area for each pen. This requires setting up a function that represents the total fencing needed and using techniques such as derivatives to find critical points that yield the minimum value.
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Intro to Applied Optimization: Maximizing Area
Area and Perimeter Formulas
Understanding the formulas for area and perimeter is crucial in this problem. The area of each rectangular pen is given as 100 m², which can be expressed as length times width (l * w). The perimeter, or the amount of fencing needed, is determined by the dimensions of the pens, and the relationship between these dimensions must be established to solve the optimization problem effectively.
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05:59Estimating the Area Under a Curve with Right Endpoints & Midpoint
Constraints
Constraints are conditions that must be satisfied in optimization problems. In this case, the area of each pen is a constraint that limits the possible dimensions of the pens. By incorporating this constraint into the optimization process, we can express one variable in terms of another, allowing us to reduce the number of variables and simplify the problem to find the optimal dimensions.
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Related Practice
Textbook Question
{Use of Tech} Basketball shot A basketball is shot with an initial velocity of v ft/s at an angle of 45° to the floor. The center of the basketball is 8 ft above the floor at a horizontal distance of 18 feet from the center of the basketball hoop when it is released. The height h (in feet) of the center of the basketball after it has traveled a horizontal distance of x feet is modeled by the function h(x) = 32x² / v² + x + 8 (see figure). <IMAGE>
b. During the flight of the basketball, show that the distance s from the center of the basketball to the front of the hoop is s = √ (x - 17.25)² + ( -(4x² / 81) + x - 2)² (Hint: The diameter of the basketball hoop is 18 inches.)
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Textbook Question
Values of related functions Suppose f is differentiable on (-∞,∞) and assume it has a local extreme value at the point x = 2, where f(2) = 0. Let g(x) = xf(x) + 1 and let h(x) = xf(x) + x +1, for all values of x.
b. Does either g or h have a local extreme value at x = 2? Explain.
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Textbook Question
{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). <IMAGE>
b. Find the time and the displacement when the object reaches its lowest point.
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Textbook Question
Cylinder in a cone A right circular cylinder is placed inside a cone of radius R and height H so that the base of the cylinder lies on the base of the cone.
b. Find the dimensions of the cylinder with maximum lateral surface area (area of the curved surface).
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Textbook Question
{Use of Tech} Demand functions and elasticity Economists use demand functions to describe how much of a commodity can be sold at varying prices. For example, the demand function D(p) = 500 - 10p says that at a price of p = 10, a quantity of D(10) = 400 units of the commodity can be sold. The elasticity E = dD/dp p/D of the demand gives the approximate percent change in the demand for every 1% change in the price. (See Section 3.6 or the Guided Project Elasticity in Economics for more on demand functions and elasticity.)
b. If the price is \$12 and increases by 4.5%, what is the approximate percent change in the demand?
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