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Turning a corner with a pole
What is the length of the longest pole that can be carried horizontally around a corner at which a corridor that is a ft wide and a corridor that is b ft wide meet at right angles?
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 4, Problema 74b
The arbelos An arbelos is the region enclosed by three mutually tangent semicircles; it is the region inside the larger semicircle and outside the two smaller semicircles (see figure). <IMAGE>
b. Show that the area of the arbelos is the area of a circle whose diameter is the distance BD in the figure.
Guida verificata passo dopo passo1
Identify the semicircles: Consider the larger semicircle with diameter AC and two smaller semicircles with diameters AB and BC, where B is the point of tangency between the two smaller semicircles.
Calculate the area of the larger semicircle: The area of a semicircle is given by \( \frac{1}{2} \pi r^2 \), where \( r \) is the radius. For the larger semicircle, the radius is \( \frac{AC}{2} \).
Calculate the area of the two smaller semicircles: Similarly, calculate the areas of the semicircles with diameters AB and BC. The radii are \( \frac{AB}{2} \) and \( \frac{BC}{2} \) respectively.
Determine the area of the arbelos: Subtract the sum of the areas of the two smaller semicircles from the area of the larger semicircle to find the area of the arbelos.
Relate the area of the arbelos to a circle: Show that this area is equivalent to the area of a circle with diameter BD. Use the formula for the area of a circle \( \pi \left( \frac{d}{2} \right)^2 \), where \( d \) is the diameter, and verify that it matches the area of the arbelos.
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Arbelos Definition
An arbelos is a geometric figure formed by three mutually tangent semicircles. It consists of one larger semicircle that encompasses two smaller semicircles, all sharing a common tangent point. Understanding the configuration of these semicircles is crucial for analyzing the area and properties of the arbelos.
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Percorso guidato
05:43
Definition of the Definite Integral
Area of Semicircles
The area of a semicircle can be calculated using the formula A = (1/2)πr², where r is the radius. In the context of the arbelos, the areas of the individual semicircles contribute to the overall area of the arbelos. Recognizing how to compute these areas is essential for deriving the area of the arbelos.
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Percorso guidato
05:59Estimating the Area Under a Curve with Right Endpoints & Midpoint
Circle Area Relation
The area of a circle is given by the formula A = π(d/2)², where d is the diameter. In the problem, it is stated that the area of the arbelos can be expressed as the area of a circle with a diameter equal to the distance BD. Understanding this relationship allows for the comparison and calculation of areas between the arbelos and the circle.
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Percorso guidato
04:16Intro To Related Rates
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Domanda del libro di testo
Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.
g(t) = 3t⁵ - 30t⁴ + 80t³ + 100
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Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.
f(x) = x⁴eˣ + x
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Domanda del libro di testo
Another pen problem A rancher is building a horse pen on the corner of her property using 1000 ft of fencing. Because of the unusual shape of her property, the pen must be built in the shape of a trapezoid (see figure). <IMAGE>
b. Suppose there is already a fence along the side of the property opposite the side of length y. Find the lengths of the sides that maximize the area of the pen, using 1000 ft of fencing.
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Domanda del libro di testo
Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.
f(x) = 2x⁴ + 8x³ + 12x² - x - 2
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Domanda del libro di testo
Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.
h(t) = 2 + cos 2t on [0,π]
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